‘Eureka!’ moments and researching Double Entry: the portrait of Fra Luca Pacioli

A journalist asked me this week if there was a ‘Eureka!’ moment while I was researching Double Entry. I said no, there wasn’t a Eureka moment, there were many Eureka moments. And several led me down sidetracks so far from my main story that they either didn’t make it into the finished book at all or have become asides.

One of these asides is the story behind the portrait of Luca Pacioli, which I discovered in ‘The Portrait of Fra Luca Pacioli‘, an article by English mathematician Nick Mackinnon published in 1993. Here Mackinnon writes about using Pacioli’s mathematics and the mathematics of the portrait in an attempt to unravel the portrait’s many mysteries. Mackinnon’s investigation was also filled with Eureka moments. He writes: ‘I have tried in this article to preserve some of the excitement of the chase, at the cost, perhaps, of a systematic presentation.’

In the portrait above, painted after he’d become a celebrity following the success of his Summa de arithmetica, geometria, proportione et proportionalita, Fra Luca Pacioli is depicted in grey Franciscan robes demonstrating a mathematical problem to a young man. Pacioli points to a diagram drawn in chalk on a slate, the edge of which is marked ‘EVCLIDES’. To his left is a large book, probably the Summa, which supports a wooden dodecahedron, one of the five Platonic solids which were revered in the Renaissance. Hanging at the top left of the painting is a crystal rhombicuboctahedron (as it was later named by Kepler), one of the 13 semi-regular solids discovered by Archimedes. All its faces (18 squares and 8 equilateral triangles) are simultaneously visible and some reflect buildings, probably the ducal palace of Urbino.

Who painted the portrait, the exquisite precision of the rhombicuboctahedron (especially compared to the crude representation of the much simpler-to-paint dodecahedron) and the identity of the young man beside Pacioli have long been the subject of debate by historians. The portrait, now in Naples’ Galleria Nazionale di Capodimonte, was dedicated to Pacioli’s patron and probable student Guidobaldo da Montefeltro, Duke of Urbino, and so it’s always been assumed that the young man in the portrait was Guidobaldo.

But as Mackinnnon points out, no amount of documentation can make ‘the handsome ambitious red-head of the Pacioli portrait into the gifted but sickly Guidobaldo’. Mackinnon was led to the ‘astonishing answer’ to the question of the young man’s identity by a remark made in Pacioli’s next book, De divina proportione (‘The divine proportion’). This led him to Pacioli’s unpublished manuscript on magic and mathematical games, De viribus quantitatis (‘On the powers of numbers’), which has languished in the Bologna University library for 500 years. De viribus includes a section on magic squares and astrology – and magic squares made Mackinnon think immediately of Albrecht Durer’s famous 1514 engraving Melencolia I, which also contains a magic square, the first 4 x 4 magic square to appear in print in Europe. (Magic squares were new on the scene and Durer’s magic square was only the third to appear in Europe, the other two being Pacioli’s and one in a manuscript now in Cracow, Jagiellonian MS 753.)

Melencolia I with a magic square at the top right-hand side

Durer's magic square

And so speculating that Durer might somehow be connected to Pacioli and therefore possibly be the young man in the portrait, Mackinnon compared a black and white portrait of Durer with the man in Pacioli’s portrait. He found they shared a similarly confident manner. He then spent another week ‘on tenterhooks’ trying to find out the colour of Durer’s hair. He discovered that Durer’s hair was ‘ginger’, like the young man’s in the portrait. And so, Mackinnon’s Eureka moment.

Durer, self-portrait at 22

Mackinnon then set off in pursuit of Durer. He discovered that Durer was in Venice in 1495, the year given on Pacioli’s portrait (which is signed JACO. BAR. VIGENNIS. P. 1495), and that Durer knew the artist who probably painted it, Jacopo de’ Barbari.

And so Mackinnon supports the majority view that it was Barbari who painted the portrait – but then makes a brand new claim for the nature of its subject. Mackinnon believes the portrait shows Pacioli teaching a real maths lesson. That here the Franciscan monk is teaching the young German painter, Durer, the geometry of ancient Greece; he is expositing Book XIV.8 of Euclid’s Elements.

Mackinnon thus argues that Pacioli’s portrait records ‘the most important meeting of the Renaissance’: the meeting of Luca Pacioli and Albrecht Durer. The young German artist would go on to become ‘one of the most original geometers in Europe’ and one of the greatest artist-mathematicians of his age.

And so Mackinnon claims that this ‘under-regarded’ Portrait of Fra Luca Pacioli captures ‘one of the greatest moments of the Renaissance, the transmission to Albrecht Durer, and hence to the world north of the Alps, of the geometry of Ancient Greece and the basis of the new art of Italy.’

Needless to say I was incredibly excited to discover this possible link between Luca Pacioli and Albrecht Durer, because it meant that Pacioli, the central character of Double Entry, was directly connected (through his knowledge of the mathematics of the new perspective painting) to all three of the great Renaissance artist-mathematicians and perspective painters: Albrecht Durer, Piero della Francesca and Leonardo da Vinci. And therefore that Pacioli held a pivotal position not just in commercial history and the history of capitalism, but also in the history of European art.

The glass rhombicuboctahedron

The other mystery that Mackinnon attempts to solve is the riddle of the exquisitely drawn glass rhombicuboctahedron, which seems to have been painted by a different hand from the one which painted the awkward dodecahedron.

Mackinnon argues that the two polyhedra were indeed painted by different hands. He believes there was ‘only one man in Europe capable of painting this object, and he is not the author of the rest of the painting!’ According to Mackinnon, in the masterly execution of the rhombicuboctahedron – which is half-filled with water and yet all its refractions and reflections are precisely captured – ‘we surely see the ineffable left hand of Leonardo da Vinci’.

Mackinnon’s argument is persuasive – especially as Leonardo would soon collaborate with Pacioli on his next project, for which Leonardo would draw a series of extraordinary illustrations of polyhedra (which will be the subject of my next Double Entry post).

Piero's Montefeltro altarpiece, with a probable portrait of Pacioli second from right

For lovers of arcane Renaissance symbolism, Mackinnon also suggests a connection between the water-filled rhombicuboctahedron in Pacioli’s portrait and the egg suspended over the Virgin Mary in Piero della Francesca’s famous Montefeltro altarpiece now in the Brera in Milan. The altarpiece was commissioned by Guidobaldo’s father Federico da Montefeltro, Duke of Urbino, to celebrate the birth of his son. (By chance it’s said to include a portrait of Luca Pacioli as Saint Peter Martyr, standing second from the right with St Peter Martyr’s distinctive head wound.) Mackinnon applies Kenneth Clark’s interpretation of the double significance of the egg – as a symbol not only of rebirth and the resurrection, but in medieval iconography as a symbol of the four elements – to the rhombicuboctahedron in the portrait.

His argument stems from the chemical theory of Plato’s Timaeus, the only Platonic dialogue known in medieval Europe, which sets out Plato’s cosmogony based on Pythagoras. According to Plato, the elements fire, earth, air and water are modelled respectively by the tetrahedron, cube, octahedron and icosahedron. The rhombicuboctahedron, with its square and triangular faces, therefore represents all four polyhedra and evokes all four elements.

According to Mackinnon, by filling it with water Leonardo (assuming he painted it) made the symbol more concrete: ‘for now the elements are represented physically too: earth by the glass, water and air contained in the glass, and fire by the bright reflections’. The fifth solid is the dodecahedron, which for Plato represented the universe. So not only does Pacioli’s portrait refer to the elements of geometry (the Platonic solids), but also to the four elements of nature. And therefore, says Mackinnon, by painting a water-filled rhombicuboctahedron the artist has ‘contrived a brilliant double reference to the Timaeus‘.

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17 Responses to ‘Eureka!’ moments and researching Double Entry: the portrait of Fra Luca Pacioli

  1. Can Jacopo de Barbari possibly be the originator of the ‘Still Life’? that seems implausible.
    Also could he have originated the ‘Aerial view’ – possible.
    Definitely take the bus to the Galleria Nazionale di Capodimonte if youre ever in Naples.
    Masaccio’s Mary Magdelena is alone worth the visit. What was Massaccio doing painting with a vanishing point in 1426, before Piero & Luca had written the books?

  2. Thanks for your comments gbrun and recommendation re Galleria Nazionale di Capodimonte which I’ll definitely follow up asap.
    I’ve always assumed Jacopo was the artist of the aerial view – if by that you’re referring to the one of Venice? And the ‘Still Life’ – why implausible?
    As for Masaccio’s vanishing point, that is a fascinating question which David Hockney takes up, among many other things, in his brilliant ‘Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters’ where he very persuasively argues that 15th century painters were using optics (mirrors and lenses) to create living projections which they then painted. Needless to say his contentions have been controversial in art history circles but they’re absolutely fascinating to me and his evidence and arguments are powerful.

  3. john says:

    Here in part response to your radionz-with-Kim Hill this morning — to discover how a name you pronounced – to my hearing at anyrate – as petroli. Had me wondering whether the same sourced petroleum 🙂

    Interesting talk you gave..(will get the book) .. allow me suggest that subsequent titles might include triple entry, perhaps even quadruple entry.. and pertaining how shadow banking and derivatives have largely taken out corporate treasuries.. worldwide. One of the reasons, no less, for why fixes are neither so simple nor fast..

    With regards

    • Thanks for your comments, John. (Nice avatar!)
      Really, it sounded like I was saying Petroli!? That’s funny – and would have been most apt if he’d also sourced petroleum. As you’ve no doubt gathered, I was actually trying to say Pacioli, pronounced PA-CHO-LI. Funny!
      And thanks for your suggested subsequent titles too – you’re so right about possible triple entry. And quadruple entry too. I don’t discuss triple entry directly in my book but allude to it when I raise questions about future of accounting/shortcomings of double entry. Although the accounting profs I’ve been talking to (Graeme Dean and Frank Clarke at University of Sydney) seem to think the corporate form itself should be questioned because it’s so difficult to account for. I think this is very sound view and am about to read their book ‘Indecent Disclosure: Gilding the Corporate Lily’.
      As for my talk being interesting – thank you for saying so. Have to say, if it was then it was all thanks to Kim Hill, she’s a brilliant, thrilling interviewer, and I’m still mulling over the many questions she raised.

  4. John Deeks says:

    Enjoyed your interview on the Kim Hill show yesterday. I have sent Radio NZ a link about The Merchant of Prato’s fourteenth century ledgers – headed up “In the Name of God and of Profit” – as a comment related to your Double Entry book. You are probably familiar with it but can find it in my blog BreathlessinOrewa.

    • Glad you enjoyed the talk, John, and thanks for mentioning the merchant of Prato – he’s fascinating and does appear in my book as one of the early adopters of double entry. Will check out your blog now, thanks so much for letting me know about it.

      • John Deeks says:

        So with Datini dead by 1410, double-entry book keeping was a long-standing practice before the 1494 publication of Pacioli’s Summa?

  5. Surely someone had painted a bowl of fruit or a dead bird before 1504?
    Even though this claim from ‘Britannica’ seems to have currency:
    “Barbari probably painted the first signed and dated (1504) pure still life (a dead partridge, gauntlets, and arrow pinned against a wall).”

  6. Oh interesting, grun77, I didn’t know Jacopo de B’s still life was supposed to be the first. But yes, people did paint bowls of fruit before 1504 (in ancient Rome at least), but I guess not ‘signed and dated’, ‘pure’ still lives. Thanks for drawing my attention to it.

  7. Yes, John, that’s right. The first surviving records of double entry we have date to around 1300, in Florence. The Venetians introduced two columns in the ledger and Luca Pacioli was the first to codify and publish in print that Venetian method, which by 1494 had probably been practised for at least a century. (And Florentines like Datini referred to the new improved double entry method as bookkeeping ‘alla Venezia’.)

    • John Deeks says:

      Thanks Jane for clarifying that. I’d better read your book. Was the “In the Name of God and of Profit” also common at the top of the ledger pages?

      • Pleasure John. And yes, amazingly, it was the done thing. I discuss it in my book. Pacioli advises merchants to thank God at the opening of every ledger. And they all did it, eg first cash book of Bank of England (1694) opens ‘Laus Deo’, praise God.

  8. Luke Elworthy says:

    Hi Jane, enjoyed your interview with Kim Hill very much, and looking forward to reading your book. Congratulations. I wanted to send you an email. Mine is lukemeredith@xtra.co.nz
    Best, Luke (Elworthy)

  9. John Logan, 33 Ashington Lane, Nashville TN 37027 says:

    The connections between Piero, Isabella, Fra Luca, Leonardo, Duerer, and de Barbari are fascinating. MacKinnon has pointed the way. If you haven’t already, read him on JSTOR. The keys to unraveling the mysteries are probably in decoding “Il Ritratto” (I’ve made some progress, and am more than willing to share), and also in Pacioli’s nearly unknown work “De ludo scachorum” which I have not yet seen.

  10. Pingback: Planet Money NPR, Luca Pacioli’s magic book De viribus quantitatis – and Double Entry USA | bookish girl

  11. Pingback: Durer’s Magic Squares | Chris Brennan

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