While I was in London I had a very entertaining conversation (from the new BBC studios) with David Kestenbaum and Jacob J Goldstein from Planet Money, National Public Radio USA, which they tell me has just gone up on the website with the excellent title ‘A Mathematician, The Last Supper and The Birth of Accounting’. Their opening line was something like ‘We’re here to talk about accounting and math – and I’m excited. Can you believe I am actually excited?’ I said I could, because much to my astonishment, I also – still – find all this talk of mathematics and accounting exciting.
But the reason I mention it is that David brought up Luca Pacioli’s book on magic, De viribus quantitatis, and asked me ‘What kind of magic tricks did he do?’ (Pacioli being the star of Double Entry.) Which reminded me that although I’ve written here about most of the edited down sections from my original manuscript of Double Entry, I haven’t yet written about one of Pacioli’s most fascinating works: his book on magic, De viribus quantitatis (‘On the power of numbers’).
De viribus quantitatis is a foundational European text of magic and number puzzles. It’s been stored for 500 years in the library of the University of Bologna and never been published – but it caused a sensation in 2007 when magician William Kalush, founder of the Conjuring Arts Research Center in New York, announced that it was being translated into English for the first time and published in 2008 to mark the 500th anniversary of its completion.
At the time The Da Vinci Code was still on the bestseller lists, so the Guardian reported the story as ‘Forgotten magic manual contains the original da Vinci code‘ and hailed it as the world’s oldest magic text. Pacioli was proclaimed as ‘Leonardo da Vinci’s best friend and teacher’. This is true: Leonardo and Pacioli were close friends and collaborators – and it is most likely that they collaborated on this particular work, because it was written when they were together at the Court of Milan, from 1496 until the French invaded in 1499.
Kalush calls De viribus ‘the first major manual that is primarily concerned with teaching how to perform magic’. He says that distinct from other sources of magic methods, which date back at least to the first century, Pacioli’s De viribus emphasises magic as performance: it ‘teaches not only the methods but also gives a glimpse into how one might perform them with an eye to entertaining an audience’. Mathematician David Singmaster considers De viribus to be ‘the foundation not only of modern magic but of numerical puzzles too’ and marvels that it is not more widely known: ‘We don’t know why, but this huge thing has been hidden away in the University of Bologna we presume since the time of Pacioli’.
De viribus is a compendium of magic, recreational mathematics and proverbs which treats maths as magic and play. Although Pacioli was a mathematician of ‘unimpeachable seriousness’, he also loved maths for the pleasure it gave him and his enthusiasm for his subject infects all he wrote.
According to his dedication, Pacioli began De viribus in 1496 and finished it 12 years later in Venice in 1508. He refers to Leonardo often in the manuscript, including making the first recorded mention of the fact that Leonardo was left handed. He also praises Leonardo’s assistance in drawing the figures for De divina proportione when ‘in happy times we two were in the same service in the marvellous city of Milan’. He then writes of ‘this our newest compendium called De viribus quantitatis‘, which he dedicated to the Marquis and Marchionese of Mantua, Gian Francesco Gonzaga and Isabella d’Este. Isabella – a leading Renaissance woman, intellect and famed chess player – gave refuge to Pacioli and Leonardo in Mantua when they were forced to flee Milan to escape the invading French army in 1499.
Like most of Pacioli’s books, De viribus was written in the vernacular, a radical thing to do in an age when Latin was the language of scholarship. It’s divided into three sections.
1. ‘Of the numerical powers’: a compilation of 81 maths games and problems, including numerical puzzles similar to Suduko. It’s a key text in the history of maths because it’s the earliest comprehensive collection of such games and puzzles, which Pacioli valued for their power to stimulate students’ mathematical ability and to encourage the use of maths in everyday life.
Of the use of maths in everyday life, Pacioli says, for example: ‘One can show how it is useful to have knowledge of the force of numbers in practice and in theory when it sometimes occurs while sailing on the sea that it is necessary to lighten the ship of either goods or of persons so that all do not perish …’!
2. ‘Of the lineal virtue and power of geometry’: a collection of geometrical puzzles and jests, which range from card tricks to how to write a sentence on the petals of a rose and how to wash your hands in molten lead: ‘Take cool well water and soak your hands for a while; then shake them, you can put them in a pan full of melted lead over a flame, and it will not cook you. It is even better if you put some ground rock alum in the water … to the uneducated … it will appear to be a miracle.’
This trick was tested by Adam Savage and Jamie Hyneman in the 2009 season finale of Mythbusters and – amazingly! – found to work.
Pacioli also explains how to make a coin ‘go up and down in a glass’, which involved magnetic powder, vinegar and a copper coin; and how to make an egg walk: empty an egg through a pinhole, fill the hole with wax and attach a long hair to the eggshell. ‘Fasten another bit of wax to the other end … placing the egg on the table, with the nail of your middle finger, pick the said wax, and by moving it here and there … it will follow. This must be done in a place not too brightly lit, with onlookers in the distance.’
Among his card tricks is this: ‘You will be able to teach the said boy, since he is closed in a room or at a distance, to guess which card some people have touched without seeing it, by way of the numbers you have agreed on with him: that is, by placing a number on the fingers and cards according to the trick, and according to the agreement made between you … since it always appears to those who do not know the way … that all these things are done by the magic art of divination.’
3. ‘Of every useful moral precept’: a collection of proverbs and verses, including 22 riddles.
Pacioli’s many references to Leonardo in De viribus suggest that they did collaborate on this manuscript as they did on De divina proportione. Especially as many of the problems it contains have been found in Leonardo’s notebooks.
Esoterica: In De viribus Pacioli gives an example of a magic square, a square of numbers in which the numbers in all the rows, columns and diagonals add up to the same number. Magic squares have an ancient lineage – they first appeared over 4,000 years ago and are thought to have originated in ancient Persia – but it was long believed that the first magic square to appear in Europe was Albrecht Durer’s in his famous 1514 engraving ‘Melencolia I‘. But Pacioli’s magic square – which appears in part I, problem 72 of De viribus – predates Durer’s but at least five years. This was one more piece in the puzzle that English mathematician Nick Mackinnon put together to solve the many mysteries surrounding the Portrait of Luca Pacioli.
In other news, I’m heading to New York in mid October for the launch of the North American edition of Double Entry, which is looking very elegant with its two columns and Canaletto painting of Venice’s Grand Canal on its cover.