I’m heading to Brisbane tomorrow to talk about double entry on ABC radio’s Conversations with Richard Fidler, so once again I’m turning my mind to the man who started it all, Luca Pacioli. One of the many fascinating asides that caught my attention while I was researching Double Entry relates to Pacioli’s De divina proportione (his mystical treatise on maths, art and God which was illustrated by Leonardo da Vinci and published in 1509) and a group of Cubist painters in Paris in 1912.
De divina proportione contains three volumes: ‘Compendio de divina proportione’ (or ‘Compendium of divine proportion’), ‘Tractato del’ architectura’ (‘Treatise on architecture’) and ‘Libellus in tres partiales tractatus divisus quinque corporum regularium et dependentium (‘Treatise on the five regular bodies’).
The first volume includes a detailed summary of the properties of the golden ratio and a study of the Platonic solids and other polyhedra, and was illustrated by Leonardo da Vinci. Pacioli finished this volume in Milan, as he notes: ‘Finished this day of December 14, at Milan in our still cloister the year 1497 under the rule of the Highest Pontiff Alexander VI in the seventh year of his pontificate.’
The second book of De divina is a treatise on proportion and its application to architecture and the structure of the human body, based on the work of the ancient Roman architect Marcus Vitruvius Pollio, known as Vitruvius. It was Vitruvius who showed that the human body could be perfectly inscribed within a circle and a square, the two most perfect geometric figures, which was famously illustrated by Leonardo.
The third and final volume of De divina is a brief work on the five regular bodies, which is essentially a verbatim translation of Piero della Francesca‘s Latin Libellus de Quinque Corporibus Regularibus. Pacioli doesn’t acknowledge Piero’s work, which has led to allegations of plagiarism against Pacioli (although as his biographer R Emmett Taylor suggests, it’s possible this volume could have been appended to Pacioli’s work later without his knowledge). Regardless of whether or not Pacioli plagiarised Piero’s work, the translation of this treatise into the vernacular and its publication in De divina made it the only one of Piero’s works to be widely available during the Renaissance.
Pacioli finished a manuscript of De divina on 14 December 1498 and dedicated it to the Duke of Milan, Ludovico Sforza. It’s now in the Bibliotheque de Geneve. Pacioli made a second version – dedicated to his patron, Galeazzo da Sanseverino, the captain-general of Milan – which is now in Milan’s Ambrosiana library. A third copy made for the gonfaloniere of Florence, Piero Soderini, is now lost.
Although the manuscript was finished in 1498, De divina wasn’t published until 1509 – probably because in October 1499 the army of Louis XII of France invaded Milan. Pacioli was dragged from his cloister by French soldiers and fled the city with Leonardo da Vinci soon after to seek refuge in Mantua in the court of Isabella d’Este. After living with Leonardo in Florence, Pacioli returned to Venice in 1508 to oversee the printing of De divina.
Four hundred years after the publication of De divina proportione in Venice, the Italian futurist painter Gino Severini rediscovered Pacioli’s book and through it the mathematics of Piero della Francesca.
In the autumn of 1912 a group of Cubist painters held an exhibition at the Boetie Gallery in Paris called ‘Section d’Or‘ – the golden section or golden ratio. According to Severini, it caused a considerable stir. The exhibition brought together for the first time all the adherents of Cubism – including Severini, Duchamp, Juan Gris, Leger, Gleizes, Metzinger and Delauney – except its two creators, Braque and Picasso, whose work was exclusively committed to the Kahnweiler Gallery.
The exhibition was conceived and named by the painter and engraver Jacques Villon. On Sunday afternoons a group of artists met at Villon’s studio near Paris to discuss problems of rhythm and proportion. Villon had developed a theory of vision based on Leonardo da Vinci, which had led him to the work of Luca Pacioli. At one of his Sunday gatherings Villon suggested that they borrow the title ‘Section d’Or’ from the treatise De divina proportion ‘by the monk Luca Pacioli and illustrated by Leonardo da Vinci himself’.
And so the Cubists adopted the golden ratio as one of the constants for the mathematical organisation of their compositions. According to art historian Maurizio Fagiolo dell’Arco, the Cubists’ interest in mathematics and its relation to art reflected ‘the profound need for order and measure that they felt more through sensibility and reason than as a result of calculation’ on the eve of the First World War.
In his autobiography The Life of a Painter Severini writes of his interest in the ancient rules of painting and his desire to use them not just superficially, as he felt Villon’s circle had been doing, but profoundly, to probe deep into their mathematics and logic. He says:
‘In many essays on painting … I confirmed that clear and precise rules had dominated artistic creativity in ancient times; this creative regimen included painting, both in the geometric and mathematical structure of the work, as well as in its technical execution.’
The story of Severini’s extensive researches into De divina proportione and the mathematics of perspective, and his application of it to his own work, is a tale of almost Pygmalion dimensions. I’ll be blogging about it here soon.
In the meantime, here are some more illustrations from De divina proportione, taken from the facsimile I saw in Sansepolcro. They are not otherwise readily available.