Abstract
The close relationship between art and science is a distinctive feature of the Renaissance, and, as we have observed again and again, Leonardo represents one of the highest points of this synthesis. The scientific knowledge that he obtained via his studies became a fundamental instrument also for his work as an artist and vice versa. As discussed in the previous chapter, it was thanks to his dear friend Luca Pacioli that Leonardo was able to fully appreciate the beauty of geometry and arithmetic. Luca Pacioli’s support was also critical, more generally, in enabling him to achieve a satisfactory knowledge of mathematics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Folicaldi F (2005) Il numero e le sue forme: storie di poliedri da Platone a Poinsot passando per Luca Pacioli. Nardini editore.
- 2.
“Come dividere un segmento in modo che il rettangolo che ha per lati l’intero segmento e la parte minore sia equivalente al quadrato che ha per lato la parte maggiore”.
- 3.
Original text: “Commo Idio propriamente non se po diffinire ne per parolle a noi intendere, così questa nostra proportione non se po mai per numero intendibile asegnare, né per quantità alcuna rationale exprimere, ma sempre fia occulta e secreta e da li mathematici chiamata irrationale.”
- 4.
Zocchi A (2005) La sezione aurea, Gli esperimenti psicologici per verificare la bellezza del rapporto aureo. Available online: www.cicap.org/new/articolo.php?id=101948. Accessed 8 August 2016.
- 5.
The Flagellation of Christ (probably 1455–1460) is an oil and tempera work realized by Piero della Francesca between 1455 and 1460 and is now in the Galleria Nazionale delle Marche in Urbino. The interpretation of the painting is still controversial because of the complexity of the iconography. Piero della Francesca used linear perspective with an unprecedented level of mastery; for the art historian Kenneth Clark, the Flagellation is “the Greatest Small Painting in the World” (Clark K (1951) Piero della Francesca. Phaidon-Press London).
- 6.
Hoge H (1997) The Golden Section Hypothesis—Its Last Funeral. Empirical Studies of the Arts 15 (2), pag 233–255.
- 7.
Three Books of Occult Philosophy Book One: A Modern Translation, written by Henry Cornelius Agrippa, Translated by Eric Purdue (2012). Renaissance Astrology Press.
- 8.
The spiral of Fibonacci is not a real mathematical spiral, because it has instead a logarithmic form.
- 9.
“Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1 Cum his itaque novem figuris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus, ut inferius demonstratur. (The nine Indian figures are these: 9 8 7 6 5 4 3 2 1. With such nine figures and the sign 0, which in Arabic is called zephiro, any number can be written, as would be later demonstrated).”
- 10.
Contin D, Odifreddi P, Pieretti A (2010) Antologia della Divina Proporzione di Luca Pacioli, Piero della Francesca e Leonardo da Vinci. Aboca Museum Edizioni.
- 11.
Luca Pacioli, Divina Proportione. Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venezia.
- 12.
Original text: “Opera a tutti glingegni perspicaci e curiosi necessaria. Ove ciascun studioso di philosophia, Prospectiva, Pictura, Scultura: Architectura, Musica e altre Mathematice: suavissima: sottile: e admirabile doctrina consequira: e delectarassi: co’ varie questione de secretissima scientia.”
- 13.
At the end of Chapter 71, Luca Pacioli annotated that the work was terminated on 14 December 1497, VII year under the pontifex Alessandro VI.
- 14.
Manuscript 170 sup. BAM.
- 15.
L. Pacioli, Divina Proportione. Reproduction of the copy at the Biblioteca Ambrosiana of Milan. Silvana Ed., Milano, (2010); L. Pacioli, Divina Proportione. Reproduction of the copy at the Bibliothèque Publique et Universitaire of Geneve. Aboca Edizioni, Sansepolcro.
- 16.
In the frontispiece of the Divina Proportione: A. Paganius Paganinus characteribus elegantissimis accuratissime imprimebat (Divine Proportion: A. Paganius Paganinus printed with very elegant and accurate characters).
- 17.
Luca Pacioli, Divina Proportione, p. 23r. Original text in vernacular: “Spero in breve sirete apieno da me satisfactit e anco con quella prometto darve piena notitia de prospectiva mediante li documenti del nostro conterraneo e contemporale di tal faculta ali nostri tempi monarcha Maestro Petro de Franceschi della qual gia feci dignitissime compendio.”
- 18.
The Divina Proportione contains altogether 87 illustrations. 23 of them show the construction of the capital letters, 59 are hollow and solid geometrical forms, 3 are illustrations of architecture, and 1 is the profile of a human head. The treatise also contains a table related to the construction of the tree of proportions. In addition, the text contains another 185 small drawings in the margins.
- 19.
Ciocci A. I poliedri regolari tra arte e geometria. Pacioli e Leonardo tra Euclide e Platone. Available online:
http://www.centrostudimariopancrazi.it/pdf/quaderno2/pagg%20,153–184.pdf. Accessed 8 August 2016
- 20.
Cornford Macdonald F (1997). Plato’s Cosmology: The Timaeus of Plato, Translated with a Running Commentary. Indianapolis: Hackett Publishing Company, Inc.
- 21.
The formula was independently discovered by Leonhard Euler (1752) and Descartes and is also known as the Euler-Descartes formula. It can be also applied to some non-convex polyhedra.
- 22.
The original of Luca Pacioli is: “El dolce frutto vago e sì diletto costrinse già i Philosophi cercare causa de che pasci l’intelletto.” Corpora ad lectorem. Divina proportione.
- 23.
Original text: “Facte et formate per quella ineffabile senistra mano, a tutte le discipline mathematici acomodatissima del principe oggi fra’ mortali, Lionardo nostro da Venci.” De viribus quantitatis. Manuscript 250 of the Biblioteca Universitaria in Bologna. Folio 237r.
- 24.
“Questi son quelli Magnanimo Duca di quali le forme materiali con asai adornezze nelle proprie mani di V.D.S. nel sublime palazzo del Reverendissimo cardinale nostro protectore Monsignor de San Pietro in vincula quando quella venne ala visitatione del Summo pontefice Innocentio 8°, negli anni della nostra salute 1489, del mese de aprile, che già sonno 5 anni elapsi. E insiemi con quelli vi foron molti altri da ditti regulari dependenti. Quali fabricai per lo Reverendo Monsegnor meser Pietro de Valetarij de Genoa, dignissimo vescovo de Carpentras, al cui obsequio alora foi deputato in casa de la felicissima memoria del R.mo Cardinale de Fois, nel palazzo ursino in campo de fiore.” L. Pacioli, Summa, second part f. 68v.
- 25.
A detailed study of the differences in the illustrations in the different manuscript versions can be found in the article: Dirk Huylebrouck. Observations about Leonardo’s drawings for Luca Pacioli. http://arxiv.org/ftp/arxiv/papers/1311/1311.2855.pdf.
- 26.
De viribus quantitatis. Folio 106r, manuscript 250, Biblioteca Universitaria, Bologna
- 27.
An Archimedean solid should not be a prism or an antiprism.
- 28.
Huylebrouck D (2012) Lost in Enumeration: Leonardo da Vinci’s Slip-Ups in Arithmetic and Mechanics. The Mathematical Intelligencer, 34: 15.
- 29.
Studies on the dodecahedron are also reported on folio 533r of the Codex Atlanticus.
- 30.
Bagatin PL (2000) Preghiere di legno. Tarsie ed. intagli di fra Giovanni da Verona. Catalogo. Centro Editoriale Toscano.
- 31.
Tormey A, Farr J (1982) Renaissance Intarsia: The Art of Geometry. Vol. 247, issue 1, pp. 136–143.
- 32.
Bagatin PL (2016) Fra Giovanni Da Verona e la Scuola Olivetana dall’Intarsio Ligneo. Antilia, Treviso.
- 33.
Swetz FJ (2013) Mathematical treasure: Wenzel Jamnitzer’s Platonic Solids. Convergence. Penn State University.
- 34.
Stoer L (2006) Geometria et perspectiva: Corpora regulata et irregulata. Manuscript Cim 103, University Library Munich. Harald Fischer Verlach.
- 35.
The Sacrament of the Last Supper is an oil on canvas painting with dimensions of 267 cm × 166.7 cm, realized by Salvador Dalì in 1955.
- 36.
Maurits Cornelis Escher (Leeuwarden, 1898–Laren, 1972) is very famous for his graphical work, which is in part inspired by geometry.
- 37.
The image can be found in the Escher’s official website: www.mcescher.com.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Innocenzi, P. (2019). The Divine Proportion. In: The Innovators Behind Leonardo. Springer, Cham. https://doi.org/10.1007/978-3-319-90449-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-90449-8_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90448-1
Online ISBN: 978-3-319-90449-8
eBook Packages: EngineeringEngineering (R0)