And now, back to my travels in Italy researching the life of Fra Luca Pacioli and Double Entry. Soon after his cryptic portrait was painted, Luca Pacioli left Venice for Milan and it was to Milan that I travelled next.
In that same city a little known engineer and master of theatrical spectaculars named Leonardo da Vinci had bought a copy of Pacioli’s mathematical encyclopaedia Summa de arithmetica, geometria, proportione et proportionalita soon after its publication in 1494.
At Leonardo’s insistence, Luca Pacioli was summonsed to the Court of Milan by its ambitious ruler, Ludovico Sforza, who was transforming his realm into a true Renaissance city: a centre of the arts and learning complete with a court of intellectuals. As part of his modernisation programme, Ludovico had recently introduced mathematics lectures – and in 1496 he invited Luca Pacioli to take up Milan’s first Chair of Mathematics.
When Pacioli met Leonardo in Ludovico’s Castello Sforza in 1496, the artist was preoccupied with mechanics, hydraulics, architecture and engineering. The two men were obsessed with arithmetic and geometry, believing, as Leonardo put it, that they embraced ‘all the things in the universe’, that without them ‘nothing can be done’.
Leonardo had recently been honoured with the first two major art projects of his career: a huge equestrian sculpture in bronze to honour Ludovico’s father (which was never cast), and a giant fresco in the church of Santa Maria delle Grazie. Leonardo had been commissioned to paint the monks’ dining room with the subject traditionally used for refectories: the Last Supper.
During his three years in Milan, Luca Pacioli collaborated with Leonardo da Vinci on several projects including his next book, De divina proportione (‘Of the divine proportion’), which he dedicated to Ludovico Sforza; De viribus quantitatis (‘On the powers of Numbers’), a compendium of magic, recreational mathematics and proverbs; and De ludo scacchorum (‘Of the game of chess’), rumoured to have been the first book on chess, which many doubted ever existed because no copy had ever been found – until 2006, when one was found among the 22,00o volumes in the library of a palazzo in northeastern Italy.
According to Pacioli, De divina proportione was inspired by the fervent discussions which regularly erupted in Ludovico Sforza’s court about the application of mathematics and natural science to art, a subject of crucial importance in the Renaissance. These debates prompted Pacioli to write a book to explain the mathematical basis of the arts for ‘everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines’.
De divina proportione is Pacioli’s most mystical book. It’s about the ‘divine proportion’, better known today as phi, or the golden mean or ratio – and its relation to the five regular solids. The golden ratio – which results when a line is divided so that the short portion is to the longer portion as the longer is to the whole – recurs with an uncanny frequency in the natural world, including in the human body (eg the navel divides the human body according to the golden ratio); in the spiral growth of shells; the proportion of a dolphin’s eye, fins and tail to its body; and the seed heads of a sunflower.
Leonardo made a set of 60 beautiful 3-D geometric drawings for De divina proportione, the first of their kind. On its publication in Venice in 1509 De divina became the best known and most successful of all Pacioli’s books. And intriguingly, 500 years after its publication, a group of Italian painters interested in the Golden Section would rediscover Pacioli’s De divina proportione and the mathematics developed by Piero della Francesca and use them to revolutionise Italian painting.
I’ll write more about De divina proportione, Pacioli and Leonardo da Vinci in the new year. In the meantime, wishing you all a very merry festive season – see you in 2012.