Mathematics, art and God: The divine proportion of Luca Pacioli with drawings by Leonardo da Vinci

The mysterious powers of mathematics and its application to art were favourite subjects in the court of Milan when Luca Pacioli and Leonardo da Vinci were there in the late 1490s – and they were the subject of Luca Pacioli’s next book. Called De divina proportione (The Divine Proportion), it explores the mathematics of the divine proportion (or golden ratio) and its relation to the heavenly spheres.

Published in Venice in 1509, De divina proportione was Pacioli’s most successful book. It was also his most mystical. The ‘divine proportion’ of the title – which clearly indicates Pacioli’s supernatural interests – is better known today as phi. This ratio results when a line is divided so that the short portion is to the longer portion as the longer portion is to the whole, and is expressed numerically as 1.61830339887…

Pacioli calls the golden ratio ‘essential, ineffable, awesome and invaluable’. In De divina he explores its relation to the five regular (or Platonic) solids: the cube, tetrahedron and octahedron (used by the Pythagoreans and probably learnt from the Egyptians), and the icosahedron and dodecahedron (developed by the Pythagoreans). The Pythagoreans had assigned each figure to an element: the cube to earth; the tetrahedron to fire; the octahedron to air; the icosahedron to water; and the dodecahedron to heavenly ether. These solids had been revered by philosophers and mathematicians since the ancient Greeks, and in the Renaissance were seen as the ‘supreme expression of the majesty of geometry’.

On the first page of De divina, Pacioli declares his desire to reveal to artists the secret of harmonic forms through the use of the divine proportion, calling his book:

‘A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.’

Pacioli says he’s included in his book all the material forms of geometric bodies which have ‘hitherto been unknown to the living’. This is true. These geometric bodies – a collection of regular and semi-regular solids – had never before been visually represented. But Pacioli was fortunate to have the ‘ineffable left hand’ of Leonardo da Vinci at his disposal – and Leonardo made a set of 60 3-dimensional illustrations which appear in two surviving manuscripts of De divina and in the printed edition of 1509.

The drawings show Leonardo’s extraordinary spatial imagination. To represent the 3-D figures in 2-D space, Leonardo devised a way of drawing them in perspective, systematically shaded as if they’re real objects, rather than geometrical diagrams, and invented a method of showing their spatial configurations in skeletal form.

Pacioli acknowledges Leonardo’s contribution with the following praise: ‘the most excellent painter in perspective, architect, musician, the man endowed with all virtues, Leonardo da Vinci who deduced and elaborated a series of diagrams of regular solids’.

De divina contains three texts: ‘Compendio de divina proportione’ (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. In the fifth chapter Pacioli discusses the divinity of numbers and explains why he’s called his book ‘divine proportion’ (the golden ratio had previously been known as ‘extreme and mean ratio’ or ‘proportion having a mean and two extremes’). Why does Pacioli call this proportion ‘divine’? Because ‘we expect in this most useful discourse God himself to come’. He gives five reasons for why he’s chosen to rename the extreme and mean ratio the divine proportion:

1. ‘That it is one and only one and not more’. That is, there’s only one value for the divine proportion and only one Christian God.

2. The geometric expression of divine proportion involves three lengths and God also contains three (the Holy Trinity of Father, Son and Holy Ghost).

3. Just as God can’t be properly defined nor understood through words, so the divine proportion can’t be designated by any intelligible number nor by any rational quantity, ‘but always remains concealed and secret, and is called irrational by mathematicians’.

4. The omnipresence and invariability of God is like the self-similarity associated with the divine proportion: its value is always the same and does not depend on the length of the line being divided or the size of the pentagon in which ratios of lengths are calculated.

5. Pacioli proposes a fifth, esoteric, quality shared by the divine proportion and the Christian God, which he derives from Plato: ‘As God has conferred being to heavenly virtue as a fifth substance, and by means of this fifth substance has extended being to the other four simple bodies or four elements (earth, water, air and fire) and through these to every other thing in nature, so in our divine proportion, following the ancient Plato in his Timaeus, we give formal being to Heaven itself by creating for it the body called the dodecahedron or the body of twelve pentagons.’

By the Middle Ages the five regular solids were relatively well known among educated Europeans because of their metaphysical significance. Euclid had demonstrated how these five solids could be constructed geometrically. Although the resulting figures were highly abstract, they did introduce into the western mathematical tradition the challenge of representing 3-dimensional figures. The work of Fibonacci continued this tradition and was picked up by Pacioli’s confrere, the painter and mathematician Piero della Francesca.

By representing the solids in three dimensions in De divina, Leonardo transformed the problem into a 3-dimensional one, an approach later taken by Durer. As Kim H. Veltman says: ‘By transforming the treatment of the regular and semi-regular polyhedrons from a two-dimensional construction problem to a three-dimensional perspectival challenge, Leonardo initiated a programme for translating the whole of Euclidean geometry into three-dimensional terms.’

And for the first time, in De divina, his colleague Luca Pacioli demonstrated the significance of these polyhedra beyond their metaphysical and geometric importance: he showed them to be the building blocks of the everyday world. For example, he discusses their application to architecture: the 26-sided rhombicuboctahedron and its stellated truncated form with 72 sides were used in the construction of the Pantheon in Rome and the Santa Maria delle Grazie in Milan.

Scientists would later discover that the presence of these forms in the everyday world goes even deeper than their use in art and architecture, to the structure of matter itself, eg the tetrahedron form of silicates and other polyhedral forms of fluorides, garnets, cuprite, etc.

500 years after the publication of Pacioli’s De divina proportione, a group of Italian painters interested in the golden ratio would rediscover his book and the mathematics developed by Piero della Francesca. Stay tuned for the story of how in 1912 Pacioli’s De divina took Paris by storm.

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18 Responses to Mathematics, art and God: The divine proportion of Luca Pacioli with drawings by Leonardo da Vinci

  1. eddy says:

    Beautiful photographs of Pacioli’s work. I had to go to the British Museum to see them with all the securty difficulties involved. You have done a wonderful service to those interested in Ph.
    See my web

  2. Thanks Eddy for the comment and the link to your great looking website. Yes, the pictures are hard to source – I’ll post some more here soon (which I took of a facsimile edition of De divina proportione).

  3. Jon Flory Schrock says:

    I am interested in any of da Vinci’s drawings of polyhedra. Short of general searches online do you have suggestions of places to look?

    • Hi Jon – Of course the best place to look is in Pacioli’s De divina proportione, but it’s not widely available, which is why I’ve put all the photographs I took of the drawings up here. The copy I read was in Sansepolcro, there’s a manuscript in Milan, one in Switzerland. Other than that I’d say the best place to start would be in a major library with the help of a librarian. All the best with your search, Jane

  4. In case you haven’t seen them, Jon, there are more pictures of Leonardo’s drawings in my post ‘De divina proportione, Luca Pacioli and Piero della Francesca’. I might have more to put up some time, but I suspect I’ve blogged them all.

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  6. Linda says:

    We are planning a trip to Italy next year, and my husband (a university lecturer in accounting) often raises Luca Pacioli. He would LOVE to see some of his manuscripts. I believe some may be found in Sansepolcro (which we may not be able to get to) anywhere else?

    • Hi Linda – Good to hear from you. Pacioli’s magnum opus, the Summa de arithmetia, geometria, proportione e proportionalita, which includes his 27-page bookkeeping treatise, is a book, not a manuscript, and the 1494 edition is held in the library of Sansepolcro and the Biblioteca Nazionale Marciana in Venice, just near Piazza San Marco (opposite the Palazzo Ducale). So if you’re going to Venice, that’s probably the best – and most beautiful – place to access it. You can email the library before you go to ask about it. His manuscript on maths and magic is in the library of Bologna University. De divine proportione is in Milan. Buon viaggio, Jane

      • Linda says:

        Thank you very much for your speedy and helpful reply!!! Would there also be a statue or two that my husband can see of Luca Pacioli. As mentioned he’s an accountant and somewhat obsessed with Luca and considers this a pilgrimage :), Linda

      • I’ve just arrived in New York so checking emails! If he loves Luca Pacioli I assume he knows about my book ‘Double Entry’ which is about Luca Pacioli, his life generally and especially his contribution to accounting? Pacioli memorabilia are mostly in Sanseplcro, his birth town, but his portrait in Naples and a painting said to contain a portrait of him is in the Brera in Milan.

      • Linda says:

        Thanks again Jane. Yes, my husband has your book (after all he is an accountant :). After all that going to Sansepolcro! Your previous correspondence identifies that the Summa is held in the library. I have had no end of trouble trying to find information on the library (opening hours and a contact email). Once again I would appreciate any assistance you may be able to provide! Warm regards Linda

      • Ah, yes. The library at Sansepolcro is very idiosyncratic and I only managed to access it through a series of fortunate meetings when I was there. I’m sorry I don’t have the magic key to gaining entrance, apart from suggesting you simply turn up when you get there and hope for the best! I suspect it would be far easier to access the 1494 edition of the Summa in Venice.

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  8. Dear Ms. Gleeson-White,
    So glad to finally find someone who has read Pacioli’s “Divine Proportion”. I searched the British Library for an English translation, and found nothing. Is there any version in English?
    Much credit is rightly given to Leonardo da Vinci and Dela Francesco, but I am convinced that Durer also met Pacioli in Italy. Jacopo di Barberi, painter of the famous portrait, shows a Durer-like person in the background, and visited Durer several times.
    I am a geometer/architect and hope to learn more. My website tries to explain (using paintings and sculpture) that the Extreme and Mean Ratio is the common denominator of all regular polyhedrons. That is; all regular 3-D figures ( 5 Platonic, and 4 Kepler-Poinsot ) can transform , or morph into any other regular polyhedron by the use of the Golden Ratio. See
    Thank you,
    Stephen Wilmoth

  9. Dear Stephen Wilmoth – thank you for your fascinating message, it’s great to hear from you. I’ve never seen a translation of the Divina although I think there may be a recent one. I’ve only seen the original and copies of the original.
    As for Durer and Pacioli, I agree, it is extremely likely that they met in Italy on one of Durer’s visits. I’ve written about it a little here and mention it in my book ‘Double Entry’ which is about Luca Pacioli and the influence of his bookkeeping treatise on the last 500 years. But it also takes in Pacioli’s other books and his connections with Renaissance art and artists, especially Leonardo and Piero but also Durer.
    Your website and work look fascinating, especially your sculpture.
    Thanks again for your message.
    all best, Jane

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