Abstract
Andrés De Mesa Gisbert proposes that the perspectives in paintings from the Duecento and Trecento were drawn arithmetically, i.e. without resorting to vanishing points. The most convincing argument for this hypothesis is that the division of two parallel lines by straight lines intersecting each other at a vanishing point (the geometric method) is equivalent to the division of these parallel lines into proportional parts (the arithmetic method). If the arithmetic method was indeed used by medieval artists, then the vanishing points exhibited ex post should be purely fortuitous. There are sound objections to this assertion, however: the lack of simple multiples and submultiples of the measurement units, the absence of proportionality ratios, the lengths of the operating series, and the correspondence of the vanishing points to visible loci in the pictures. The application of optics and the geometric method is a more probative thesis, although it does not imply that painters were using concepts of linear perspective, which would have been an anachronism.
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- 1.
Judith V. Field, “Alberti, the abacus and Piero della Francesca’s proof of perspective,” Renaissance Studies 11/2 (1997): 61–88.
- 2.
Luciano Bellosi, Giovanna Ragionieri, “Giotto e le storie di San Francesco nella basilica superiore di Assisi,” in Assisi anno 1300, edited by S. Brufani and E. Menestò, Assisi, Edizioni Porziuncola, 2002, pp. 455–473.
- 3.
Angiola Maria Romanini, “Arnolfo pittore: pittura e spazio virtuale nel cantiere gotico,” Arte medievale 11 (1997): 3–33.
- 4.
Hayden B.J. Maginnis, Andrew Ladis, “Assisi today: the upper church,” Source 18 (1998): 1–6.
- 5.
“Having finished these works [in Arezzo, Giotto] betook himself to Assisi, a city of Umbria, being called thither by Fra Giovanni di Muro della Marca, then General of the Friars of St. Francis, where in the upper church he painted a fresco, under the gallery that crosses the windows, on both sides of the church, thirty-two scenes from the life and acts of St. Francis/ Finite queste cose [in Arezzo, Giotto] si condusse in Ascesi città dell’Umbria, essendovi chiamato da fra’ Giovanni di Muro della Marca, allora Generale de’ frati di san Francesco, dove nella chiesa di sopra dipinse a fresco, sotto il corridore che attraversa le finestre, dai due lati della chiesa, trentadue storie della vita e fatti di San Francesco,” Giorgio Vasari, Vite de’ più eccellenti Pittori Scultori e Architettori, edited by R. Bertanini and P. Barocchi, Florence, 1967, II, p. 100.
- 6.
Elvio Lunghi, La Basilica di San Francesco di Assisi, Antella, 1996, pp. 66–67.
- 7.
Bellosi and Ragionieri, “Giotto e le storie di San Francesco,” op. cit.; Donal Cooper, Janet Robson, “Pope Nicholas IV and the upper church at Assisi,” Apollo 157 (2003): 31–35.
- 8.
Vasari, Vite, op. cit., p. 199; Cennino Cennini, Il Libro dell’arte, edited by Gaetano and Carlo Milanesi, Florence, 1859, p. 60.
- 9.
Giuseppe Basile, ed., Pittura a fresco. Tecniche esecutive, cause di degrado, restauro, Florence, 1989; Bruno Zanardi, Chiara Frugoni and Federico Zeri, Il Cantiere di Giotto, Milan, 1996.
- 10.
Bernhard Degenhart, Annegrit Schmitt, Einleitung a Corpus der Italienischen Zeichnungen 1300–1450, I–1. Sud- und Mittelitalien, Berlin, 1968, p. xix.
- 11.
“Immediately above the aureola of Francis, one can see a very small, completely indecipherable trace of a drawing in red sinopia. But, above and beyond this infinitesimal fragment of proof is the vast complexity of the iconographic cycle that precludes the hypothesis of an execution a fresco of the scenes without an extremely detailed preparatory drawing on paper (or parchment) that could be transferred directly to the arriccio [the rough underlayer of plaster] in the form of a sinopia… These material data as well confirm the fact that there must have been quite detailed preliminary planning for the Franciscan fresco cycle, first with drawings on paper (or parchment) and then on the arriccio/ Subito sopra l’aureola di Francesco è visibile una piccolissima traccia, del tutto indecifrabile, d’un disegno in rosso sinopia. Ma, al di là di questa minima prova è l’enorme complessità iconografica del ciclo a rendere impossibile l’ipotesi di una esecuzione a fresco delle scene in assenza d’un detagliatissimo progetto su carta (o pergamena) da riportare al vere sull’arriccio in forma di sinopia… Questi dati materiali impongono di nuovo di dar per certo un assai dettagliato lavoro di progettazione del ciclo francescano, prima con disegni su carta (o pergamena) e poi sull’arriccio,” Zanardi, Il Cantiere di Giotto, pp. 24, 32.
- 12.
Zanardi, Il Cantiere di Giotto, pp. 24–32.
- 13.
Zanardi, Il Cantiere di Giotto, p. 29 (Figs. 18, 19, 20), p. 31 (Figs. 22, 23, 24).
- 14.
John White, The Birth and Rebirth of Pictorial Space, London, 1967, p. 32.
- 15.
See not only Panofsky’s groundbreaking “Die Perspective als symbolische Form,” Vorträge der Bibliothek Warburg, 4 (1924/5): 258–331, Perspective as Symbolic Form, Zone Books, New York, 1991, but also Guido J. Kern, “Die Anfänge der zentralperspektivischen Konstruktion in der italianischen Malerei des 14. Jahrhunderts,” Mitteilungen des Kunsthistorischen Instituts in Florenz 2 (1913): 39–65, and Decio Gioseffi, Perspectiva artificialis. Per la storia della prospettiva, spigolature e appunti, Trieste, 1957. This thesis was amended by Rocco Sinisgalli, Per la storia della prospettiva, 1405–1605, Rome, 1978.
- 16.
Panofsky, Perspective as Symbolic Form, p. 57.
- 17.
Andrés de Mesa Gisbert, “El ‘fantasma’ del punto de fuga en los estudios sobre la sistematización geometrica de la pintura del siglo XIV,” D’Art 15 (1989): 29–50. The author has since specialized in architectural surveys.
- 18.
“Si disponemos dos rectas paralelas con cualquier distancia entre sí, y luego de dividir una de ellas en un número cualquiera de partes lo hacemos en forma similar sobre la segunda paralela, guardando exactamente las mismas proporciones con las que se lo ha hecho inicialmente, al unir los puntos correspondientes con líneas rectas, en su prolongación obtendremos la convergencia de todas ellas sobre un solo y único punto sin necesidad de haber operado con él,” De Mesa Gisbert, “El ‘fantasma’ del punto de fuga,” p. 33 (italics mine).
- 19.
De Mesa Gisbert, “El ‘fantasma’ del punto de fuga,” pp. 33–34. This idea is discussed by Ian Verstegen, “Viewer, Viewpoint, and Space in the Legend of St. Francis: A Viennese-Structural Reading,” preprint 2011.
- 20.
The arithmetical method supports the idea that perspective was a Renaissance invention. De Mesa speaks of Brunelleschi’s contribution in “El ‘fantasma’ del punto de fuga,” p. 35. For a critique, see D. Raynaud, L’Hypothèse d’Oxford, Paris, 1998, pp. 4–9 and 132–150.
- 21.
A similar example may be found in Lescot’s façade for the Louvre, whose measurements are multiples of the pied du Roi (326.6 mm) used in Paris ca. 1546, Jean-Paul Saint Aubin, “Photogrammétrie et étude des ordres: le Louvre de Lescot,” in L’Emploi des ordres à la Renaissance, ed. Jean Guillaume, Actes du colloque de Tours (9–14 juin 1986), Paris, 1992, pp. 219–226.
- 22.
Konrad Hecht, “Maßverhältnisse und Maße der Cappella Pazzi,” Architectura 6 (1976): 148–174.
- 23.
Jean Guillaume, “Désaccord parfait: ordres et mesures dans la chapelle des Pazzi,” Annali di Architettura 2 (1991): 9–23.
- 24.
“Brachium continet 12 vntias,” “Pes palmorum quattuor, pollicum seu vnciarum duodecim, digitorum vero sexdecim.” We leave aside the quattrino, whose narrow step (9.98 mm) is not sufficiently differential. Ronald E. Zupko, Italian Weights and Measures from the Middle Ages to the Nineteenth Century, Philadelphia, The American Philosophical Society, 1981, pp. 47–48 (braccio), 197 (piede).
- 25.
We have relied on the photographic survey by Zanardi, Frugoni and Zeri in Il Cantiere di Giotto, p. 128. The survey scale can be deduced from the dimensions of the fresco (363 × 357 cm), the height of the standing figure of St Francis (122 cm), the height of the other Figs. (122, 111, 123 cm), and the diameter of the saint’s halo (35.5 cm). We systematically checked the parallelism and absence of distorsion in the fresco, following Raynaud, “La théorie des erreurs et son application à la reconstruction des tracés perspectifs,” see Appendix A.
- 26.
Zanardi, Il Cantiere di Giotto, p. 242.
- 27.
Zanardi, Il Cantiere di Giotto, p. 332. The Recovery of the Wounded Man is one of the earliest works to present a correct foreshortening of the intervals, but it does not represent a case of linear perspective because the correct perspective is limited to the coffered ceiling. (1) The side ceilings are depicted in oblique perspective, while the main ceiling is depicted in a central perspective. (2) The horizon is situated 722 mm above the level of the eye, with which it should instead coincide. (3) There is a lack of consistency in the foreshortening. The most remote horizontal line of the ceiling produces an interval that is as high as the one immediately preceding it, probably because of some confusion between this line and the one that marks the boundary of the coffered space. In perspective, however, two equal intervals ought to be of different heights. (4) The fresco displays some minor errors of drawing. For example, the axis of the ceiling is shifted 12 mm to the right compared to the axis of the composition.
- 28.
These braccio values are given, for instance, by Konrad Hecht, “Maßverhältnisse und Maße der Cappella Pazzi”; Leonardo Benevolo, Stefano Chieffi e Giulio Mezzetti, “Indagine sul S. Spirito di Brunelleschi,” Istituto di Storia dell’Architettura. Quaderni 85/90 (1968): 1–52; and Christoph L. Frommel, Der Römische Palastbau der Hochrenaissance, 3 Bde, Tübingen, 1973.
- 29.
Zupko, Italian Weights and Measures, p. 46.
- 30.
De Mesa, “El ‘fantasma’ del punto de fuga,” pp. 34 (Fig. 8). The concept of proportional ratios was repeated, without success, by Pietro Roccasecca, “La prospettiva lineare nel Quattrocento: dalle proporzioni continuata e ordinata alla proporzione degradatta,” S. Rommevaux et al., eds., Proportions. Science—Musique—Peinture & Architecture, Turnhout, 2011, pp. 277–297.
- 31.
“Hic essent nonnulli qui unam ab divisa aequedistantem lineam intra quadrangulum ducerent, spatiumque, quod inter utrasque lineas adsit, in tres partes dividerent. Tum huic secundae aequedistanti lineae aliam item aequedistantem hac lege adderent, ut spatium quod inter primam divisam et secundam aequedistantem lineam est, in tres partes divisum una parte sui excedat spatium id quod sit inter secundam et tertiam lineam, ac deinceps reliquas lineas adderent ut semper sequens inter lineas esset spatium ad antecedens, ut verbo mathematicorum loquar, superbipartiens…” Leon Battista Alberti, De la peinture/De pictura (1435), eds. Schefer and Deswarte-Rosa, Paris, 1992, pp. 116–117, commentary p. 242.
- 32.
The most influential texts were those by Boethius, De Institutione arithmetica, ed. by J.-Y. Guillaumin, Paris, 1995; Jordanus de Nemore, De Elementis arithmetice artis. A Medieval Treatise on Number Theory, ed. by Hubert L.L. Busard, Stuttgart, 1991; and Hubert L.L. Busard, “Die Traktate ‘De proportionibus’ von Jordanus Nemorarius und Campanus,” Centaurus 15 (1971): 193–227. Correct definitions can also be found in less well-known treatises, such as an anonymous Tractatus proportionum: “The first species of the superpartiens genus is the superbipartiens proportion, which is produced when the greatest number contains the entire smallest plus two parts, such as 5 to 3, 7 to 5. The second above-mentioned species is the supertripartiens proportion, which is produced when the greatest number contains the entire smallest plus three parts, such as 7 to 4, 11 to 8, etc./ Prima species superpartienti generis est proportio superbipartiens, que fit quando maior numerus continet totum minorem et insuper eius duas partes, ut 5 ad 3, 7 ad 5. Secunda species supradicti generis est proportion supertripartiens, que fit quando maior numerus continet in se totum minorem in se et insuper eius tres partes, ut sunt 7 ad 4, 11 ad 8, etc.,” Saint-Dié, Bibliothèque municipale, MS. 42, fol. 119r.
- 33.
Supertripartiens, superquadripartiens, and superquinquepartiens proportions were formed on the same pattern.
- 34.
To match the observed series to the superbipartiens series, a non-integer q would be required (the optimal matching would be for q = 2.3 that provides the terms 43.45, 49.97, 57.47, 66.09, 76.00). This, however, is impossible by definition.
- 35.
The most popular text was Adelard of Bath’s version of Euclid’s Elements; for the purposes of our discussion see H.L.L. Busard (ed.), The First Latin Translation of Euclid’s Elements commonly ascribed to Adelard of Bath, Books I–VIII and Books X.36–XV.2, Toronto, The Pontifical Institute of Medieval Studies, 1983. For a mathematical commentary, see Euclid, Les Éléments, ed. by Bernard Vitrac, 4 vols., Paris, PUF, 1990–2001; also of interest is the commentary on Euclid’s Elements by al-Nayrīzī, Anaritii in decem libros priores Elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis, edidit M. Curze, Leipzig, 1899.
- 36.
See the treatises by Boethius (Menso Folkerts, « Boetius » Geometrie II: Ein mathematisches Lehrbuch des Mittelalters, Wiesbaden, 1970); Abraham bar Ḥiyya (Maximilian Curze, Der Liber Embadorum des Abraham bar Chijja Savasorda in der Übersetzung des Plato von Tivoli, Leipzig, 1902); the anonymous Artis cuiuslibet consummation (Stephen K. Victor, Practical Geometry in the High Middle Ages, Philadelphia, 1979); Leonardo Fibonacci (Baldassare Boncompagni, Leonardi Pisani. Practica geometriae, Roma, 1862); Abū al-Wafā’ al-Būzjānī (Kitāb fī mā yahtāju ilayhi al-sāni‘ min a‘māl al-handasa, ed. by Svetlana Krasnova, “Abu-l-Vafa al-Buzdjani, Kniga o tom, chto neobhodimo remeslenniku iz geometricheskih postroenij” in Fiziko-Matematicheskie Nauki v Stranah Vostoka 1 (1966): 56–130 [text], 131–140 [commentary]); and Nicolas Chuquet (La Géométrie. Première géométrie algébrique en langue française (1484), ed. Hervé L’Huillier, Paris, 1979).
- 37.
See Appendix A.
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Raynaud, D. (2016). De Mesa’s Hypothesis Regarding the Arithmetic Construction of Perspective. In: Studies on Binocular Vision. Archimedes, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-42721-8_11
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