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Mathematics and the Art and Science of Building Medieval Cathedrals

  • Josep Lluis i GinovartEmail author
Living reference work entry

Abstract

The Cathedral of Tortosa (1345) is one of the most important Gothic buildings of Catalonia (Spain). Its relevance is partly due to the large volume of documents that survive from the era of its construction. One of the most important of these documents is the Guarc’s parchment (c.1345–1380), which contains the drawing of the plan of a cathedral that was never built but is the oldest drawing of this type in Spain and offers us the information about the construction of geometric figures, such as the heptagon and the octagon in medieval times. The study of the Gothic layout through its imprints enables us to establish the geometric and arithmetic knowledge of the agents involved in the design and construction of the cathedral. This chapter provides a background to the use of mathematics in Gothic architecture, by way of a detailed study of the Cathedral of Tortosa.

Keywords

Medieval geometry Medieval drawing Heptagon Octagon Dome Tortosa Cathedral 

Abbreviations

ACTo

Arxiu Capitular Tortosa (Chapter Archives of Tortosa)

AHCTE

Arxiu Històric Comarcal Terres Ebre

FBMPM

Fundación Bertomeu March Palma Mallorca

Notes

Acknowledgments

Members group research Patriarq: PhD Agustí Costa Jover, PhD Sergio Coll Pla, and Arq. Mónica López Piquer.

References

  1. Aghayani-Chavoshi (2010) Ketâb al-nejârat. Sur ce qui est indispensable aux artisans dans les constructions géométriques. Written Heritage Re- search Centre & Institut, TehranGoogle Scholar
  2. Alonso M (1955) Domingo Gundisalvo. De Scientiis. Compilación a base principalmente de la de Al-Farabi. Consejo Superior de Investigaciones Científicas, MadridGoogle Scholar
  3. Arfe J (1585) Ioan de Arphe y Villafañe (1535-1603). De varia commensuracion para la escultura y architectura. Andrea Pescioni y Juan de León, SevillaGoogle Scholar
  4. Armisen-Marchetti M (2001) Macrobe. Commentaire au Songe de Scipion. Livre I. Les Belles Lettres, ParisGoogle Scholar
  5. Armisen-Marchetti M (2003) Macrobe. Commentaire au Songe de Scipion. Livre II. Les Belles Letres, ParisGoogle Scholar
  6. Baron R (1956) Hugonis de Sancto Victore Practica geometriae. Osiris Año 1956:176–224MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bayerri E (1962) Los Códex Medievales de la Catedral de Tortosa. Novísimo inventario descriptivo. Talleres Gráficos Algueró y Baiges, TortosaGoogle Scholar
  8. Beaujouan G (1963) Calcul d’expert, en 1391, sur le chantier du Dôme de Milan. In: Le Moyen Age 79. Livraire Jubilaire, Bruxelles, p 555–563Google Scholar
  9. Beaujouan G (1975) Réflexions sur les rapports entre théorie et pratique au Moyen Âge. In: Murdoch E, Sylla ED (eds) The cultural context of medieval learning. J. Reidel Publishing Company, Dordrech, pp 437–484CrossRefGoogle Scholar
  10. Boncompagni B (1854) Intorno ad alcune opere di Leonardo Pisano matematico del secolo decimoterzo. Tipografia delle Belle Arti, RomaGoogle Scholar
  11. Bouvelles C (1510) Liber de intellectu, Liber de sensibus, Libellus de Nihilo, Ars oppositorum, Liber de generatione. Liber de Sapiente, Liber de duodecim numeris… Henri Estienne, ParisGoogle Scholar
  12. Bouvelles C (1542) Livre singulier et utile, touchant l’art praticque de geometrie, composé nouvellement en françoys, par maistre Charles de Bouvelles. Regnaud Chaudière et Claude, ParisGoogle Scholar
  13. Bruins EM (1964) Codex Constantinopolitanus Palatii Veteris no. I, Janus Suppl. 2, 3 vols. Brill, LeidenGoogle Scholar
  14. Bubnov N (1899) Gerberti postea Silvestri II papae opera mathematica. Friedlände, Berlin, pp 972–1003Google Scholar
  15. Bucher F (1979) Architector. The lodge books and sketchbooks of medieval architects. V.1. Abaris Books, New YorkGoogle Scholar
  16. Burnett C (1996) Algorismi vel helcep decentior Est diligentia: the arithmetic of Adelard of Bath and his circle. In: Folkerts M (ed) Mathematische Probleme im Mittelalter: der lateinische und arabische Sprachbereich. Otto Harrassowitz, Wiesbaden, pp 221–331Google Scholar
  17. Burnett C (2006) The semantics of Indian numerals in Arabic, Greek and Latin. J Indian Philos 34:15–30zbMATHCrossRefGoogle Scholar
  18. Busard HLL (1965) The Practica geometriae of Dominicus de Clavasio. Arch Hist Exact Sci 2:520–575MathSciNetzbMATHCrossRefGoogle Scholar
  19. Cantor M (1907) Vorlesungen Uber Geschichte Der Mathematik, V1.2. B.G. Teubner, LeipzigGoogle Scholar
  20. Capella M (2001) Le Nozze di Filologia e Mercurio. ed. and trad. Bompiani, MilanoGoogle Scholar
  21. Cornford FM (1937) Plato’s cosmology. The Timaeus of Plato. Kegan Paul, Trench, Trubner & Co, LondonGoogle Scholar
  22. Curtze M (1887) Jordani Nemorarii Geometria vel de triangulis libri IV. Mitteilungen des Coppernicus-Vereins zu Thorn 6/1887Google Scholar
  23. Curtze M (1897) Petri Philomeni de Dacia in algorismun vulgarem Johannis de Sacrobosco. Petri Philomeni de Dacia. A. F. Hoest & Fil, CopenhagenGoogle Scholar
  24. Dürer A (1525) Underweysung der Messung, mit dem Zirckel und Richtscheyt: in Linien Ebnen vo gantzen Corporen. Hieronymus Andreae, NürembergGoogle Scholar
  25. Euclides (1576) Los seis libros primeros de geometría de Evclides: traduzidos en lengua española por Rodrigo Çamorano Astrólogo y Mathemático y Cathedrático de Cosmographía por su Magestad en la casa de la Contratacion de Sevilla. Casa de Alfonso de la Barrera, SevillaGoogle Scholar
  26. Friedlein G (1867) De institutione arithmetica libri duo. De institutione Musica libri quinque. In: Aedibus BG (ed) Accedit geometria quae fertur Boetii. Teubneri, LeipzigGoogle Scholar
  27. Font A (1891) La catedral de Barcelona. Ligeras consideraciones sobre su belleza arquitectónica. Imprenta y Litografía en Henrich y Cª, en comandita, BarcelonaGoogle Scholar
  28. Frederick AH (1991) Practical geometry (Practica Geometriae). Attributed to Hugh of St. Victor translated from the Latin with an introduction, notes and appendices. Marquette University Press, MilwaukeeGoogle Scholar
  29. García S (1991). Compendio de arquitectura y simetría de los templos conforme a la medida de cuerpo humano con algunas demostraciones de geometría: año de 1681: recoxido de diversos autores, naturales y extranjeros. Fascímil 1683. Colegio Oficial Arquitectos en Valladolid, ValladolidGoogle Scholar
  30. Gauss CF (1801) Disquisitiones Arithmeticae. Auctore D. Carolo Federico Gauss. In: Commissis apud Gerh. Fleischer Jun, LipsiaeGoogle Scholar
  31. González Á (1932) Alfarabi. Catálogo de las ciencias. Facultad de Filosofia y Letras, MadridGoogle Scholar
  32. Halliwell JO (ed) (1841) Rara mathematica; or, a collection of treatises on the mathematics and subjects connected with them, from ancient inedited manuscripts. Samuel Maynard, LondonGoogle Scholar
  33. Heath TL (1908) V2 the thirteen books of Euclid’s elements. University Press, CambridgeGoogle Scholar
  34. Heiberg JL (1914) Heronis quae feruntur Stereometrica et De mensuris. Heronis Alexandrini Opera quae supersunt omnia. Teubner, LeipzigGoogle Scholar
  35. Herrera J (1988) Sobre la figura cúbica. Universidad de Cantabria, SantanderGoogle Scholar
  36. Hirschvogel A (1543) Ein a igentliche vnd grundtliche anweysung in die Geometria. Berg und Neuber. Vom Berg und Neuber, NurembergGoogle Scholar
  37. Hoffstadt F (1847) Principes du Style Gothique. Exposés d’après des documents authentiques du moyen Âge avec 40 planches in-folio. A l’usage des artistes et des ouvriers. A. Franck et Victor Didron, ParisGoogle Scholar
  38. Hogendijk JP (1984) Greek and Arabic constructions of de regular heptagon. Arch Hist Exact Sci 30(3–4):197–330MathSciNetzbMATHCrossRefGoogle Scholar
  39. Høyrup J (2006) The rare traces of constructional procedures in practical geometries. Creating shapes. A workshop held at the Max-Planck-Institut für Wissenschaftsgeschichte, 7–9 Dec 2006, BerlinGoogle Scholar
  40. Høyrup J (2009) The rare traces of constructional procedures in ‘practical geometries’. In: Nowacki H, Lefèvre W (ed) Creating shapes in civil and naval architecture. Brill, Leiden & Boston, pp 367–377Google Scholar
  41. Kepler J (1864) V5. Joannis Kepleri Astronomi Opera Omnia. Volumen Quintum. Edidit Dr. Ch. Frisch. Heyder & Zimmer, Francofurti A.M. et ErlangaeGoogle Scholar
  42. Knorr WR (1998) On Archimedes “Construction of the Regular Heptagon”. Centaurus 32:257–271MathSciNetzbMATHCrossRefGoogle Scholar
  43. Levey M (1952) The encyclopedia of Abraham Savasorda: a departure in mathematical methodology. Isis 43(3):257–264zbMATHCrossRefGoogle Scholar
  44. López de Arenas D (1633) Breve Compendio de la Carpintería de lo blanco y tratado de alarifes, con la conclusión de la regla de Nicolás Tartaglia, y otras cosas tocantes son la Iometria, y puntas de compás. Luis Estupiñan, SevillaGoogle Scholar
  45. Lorenzo de San Nicolás (1633) Arte y uso de la Arquitectura. Dirigida Al Smo Patriarca S. Ioseph. Compuesto por Fr. Laurencio de S Nicolas, Agustino Descalço, Maestro de obras. MadridGoogle Scholar
  46. Lorenzo San Nicolás (1639) Arte y Uso de Architectura. Dirigida al Smo. Patriarca S. Ioseph. Compuesto por Fr. Laurencio de S. Nicolás, Agustino Descalco, Maestro de Obras S.l: s.n., s.aGoogle Scholar
  47. Lundy M (1998) Sacred geometry. Wooden Books/Walker & Company, New YorkGoogle Scholar
  48. Menéndez G (1959) Los llamados numerales árabes en occidente. Boletín de la Real Academia de la Historia 145:45–116Google Scholar
  49. Millàs JM (1931) Llibre de geometria: Hibbur hameixihà uehatixbòret. Alpha, BarcelonaGoogle Scholar
  50. Moreschini C (2003) Calcidius. Commentario al Timeo di Platone. Bonpiani, MilanoGoogle Scholar
  51. Muñoz I (1683) Manifiesto geométrico, plus ultra de la Geometría práctica, adición al IV libro de los elementos de Euclides. Francisco Foppens, BrusselsGoogle Scholar
  52. Navascués P (1974) El libro de arquitectura de Hernán Ruiz, el Joven. ETSAM, MadridGoogle Scholar
  53. Nuere E (2001) Nuevo tratado de la carpintería de lo blanco y la verdadera historia de Enrique Garavato carpintero de lo blanco y maestro del oficio. Editorial Munilla-Lería, MadridGoogle Scholar
  54. Olleris A (1867) Oeuvres de Gerbert, Pape sous le nom de Sylvetres II collationnées sur les manuscrits, précédées de sa biographie, suivies de notes critiques & historiques. F. Thibaud, Impr.-Libr.-Éditeur, Clermond-FerrandGoogle Scholar
  55. Özdural A (2002) The church of St. George of the Latins in famagusta: a case study on medieval metrology and design techniques. In: Wu N (ed) Ad quadratum. Ashgate, BurlingtonGoogle Scholar
  56. Pacioli L (1494) Summa de Aritmética, Geometría, Proportioni et Proporcionalità. Paganino de Paganini, VeneziaGoogle Scholar
  57. Panofsky E (1951) Gothic architecture and scholasticism. Archabbey Press, PennsylvaniaGoogle Scholar
  58. Pez B (1853) SS Silvester II. Geometria [0999–1003]. Patrologia Latina Vol. 139. MPL139, Col. 0091 – 0152C. Apud Garnier Fratres, ParisGoogle Scholar
  59. Portet P (1995) Bertrand Boysset, arpenteur arlésien de la fin du IVe. Siècle. 1355–1415. Toulouse: Université de Toulouse II. Le Mirail. Thèse d’habilitationGoogle Scholar
  60. Puig i Cadafalch J (1923) El problema de la transformació de la catedral del nord importada a Catalunya. Contribició a l’estudi de l’arquitectura gòtica meridional. In: Miscel·lania Prat de la Riba. Institut Estudis Catalanas, SeparataGoogle Scholar
  61. Rashed R (1976) Ibn al-Haytham’s construction of the regular heptagon. J Hist Arab Sci 3(2):387–409MathSciNetGoogle Scholar
  62. Raynaud D (2008) Geometric and arithmetical methods in early medieval perspective. Physis; rivista internazionale di storia della scienza 45:29–55MathSciNetzbMATHGoogle Scholar
  63. Recht R (1980) Le traité de Géométrie de Mathieu Roriczer. Les bâtisseurs du Moyen-Âge Histoire et Archéologie, 24–25Google Scholar
  64. Reynolds M (2001) From pentagon to heptagon: a discovery on the generation of the regular heptagon from the equilateral triangle and pentagon. Nexus Netw J 3(2):139–145CrossRefGoogle Scholar
  65. Reynolds M (2008) The octagon in leonardo’s drawings. Nexus Netw J Architect Math 10(1):51–76MathSciNetzbMATHCrossRefGoogle Scholar
  66. Roriczer M (1999) Das Büchlein von der Fialen Gerechtigheit (fak. Regensburg 1486) und Die Geometria Deutsch (fak. Regensburg un 1487/88). Hürtgenwald Guido Pressler, RegensburgGoogle Scholar
  67. Saidan AS (1974) The Arithmetic of Abū’l-Wafā’. Isis 65(3):367–375MathSciNetzbMATHCrossRefGoogle Scholar
  68. Sarrade MT (1986) Sur les connaissances mathématiques des bâtisseurs de cathédrales. Librairie du Compagnonnage, ParisGoogle Scholar
  69. Schoene H (1903) Heronis Alexandrini opera quae supersunt omnia. Vol III: Rationes dimetiendi et commentatio dioptrica. Teubner, Leipzig.Google Scholar
  70. Sebastián S (1994) Mensaje Simbólico del Arte Medieval. Arquitectura, Liturgia e Iconografía. Ediciones Encuentro, MadridGoogle Scholar
  71. Serlio S (1545) Il Primo libro d’architettura di Sebastiano Serlio, bolognese. Le premier libre d’Architecture de Sebastiano Serlio, Bolognoi, mis en lange Francoyse par Iehan Martin. Jean Barbé, ParísGoogle Scholar
  72. Shelby LR (1977) Gothic design techniques. The fifteenth-century design booklets of Mathes Roriczer and Hanns Schumuttermayer. Southern Illinois University Press, CarbondaleGoogle Scholar
  73. Silberberg M (1895) Sefer Ha-Mispar. Das Buch der Zahl, ein habräisch-Werk arithmetisches Werk des R. Abraham ibn Esra. J. Kauffmann, Frankfurt am MainGoogle Scholar
  74. Simson OG (1956) The Gothic Cathedral: the origins of gothic architecture and the medieval concept of Order Harper & Row. Evanston, New YorkGoogle Scholar
  75. Stevin S (1605) Tomus Secundus Mathematicarum hypomnematun de Geometriae Praxi. Ex Officina Ioannis Patii, Academiae Typographi, Lugodini BatavorumGoogle Scholar
  76. Street GE (1926) In: Murray J (ed) Some account of gothic architecture in Spain. Albemarle Street, LondonGoogle Scholar
  77. Suter H (1922) Das buch der geometrischen Konstruktionen. Suter H. Das Buch der geometrischen Konstruktionen des Abu’l Wefa. Abhandlungen zur Geschichte der Naturwissenschaften und Medizin. 4, p 94–109Google Scholar
  78. Svenshon H (2010) ‘Schlag’ nach bei Hheron. Der Turm der Winde im Sspiegel antike Vvermessungslehre’. Koldewey-Gesellschaft Vereinigung für Baugeschichtliche Fforschung E.V. Bericht über die 45. Tagung für Ausgrabungswissenschaft und Bauforschung. Von 30 april bis 4 maig 2008 in Regensburg. Klaus Tragar, AugsburgGoogle Scholar
  79. Toomer GJ (1984) Ptolemy’s Almagest. Translated and annotated by G.J. Toomer. With a foreword by Owen Gingerich. Duckworth, LondonGoogle Scholar
  80. Valla G (1501) De expetendis et fugiendis rebus Impr. Aldo Manuce, VenetiisGoogle Scholar
  81. Viète F (1696) Francisci Vietae Opera Mathematica. In unum Volumen congesta ac recognita. Ex Officina Bonaventurae & Abrahami Elzeviriorum, Lugduni BatavorumGoogle Scholar
  82. Waszink JH (ed) (1975) Plato Latinus. IV. Timaeus, a Calcidio translatus commentarioque instructus. The Warburg Institute, LondonGoogle Scholar
  83. Watts CM (1996) The Square and the Roman House: Architecture and Decoration at Pompeii and Herculanum. In: Nexus Network Journal: Architecture and Mathematics. Edizioni dell’Erba, Firenze, pp 167–182Google Scholar
  84. Willis J (ed) (1970) Macrobius. Commentarii In Somnium Scipionis (Macrobii Opera Vol II). Teubner, LeipzigGoogle Scholar
  85. Willis J (ed) (1983) Martianus capella. Teubner, LeipzigGoogle Scholar
  86. Woepcke MF (1855) Recherches sur l’historie des Sciences mathématiques chez les orientaux, dápres des traités inedits Arabes et Persans. Deuxième siecle. Analyse et extrait d’un recueil de constructions géométriques par Abûl Wafa. Journal asiatique. Fevier-Avril. V: 218–256:309–359Google Scholar
  87. Worringer W (1911) Formprobleme der Gotik. Piper Verlag, MünchenGoogle Scholar

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Authors and Affiliations

  1. 1.Universitat Internacional de CatalunyaBarcelonaSpain

Section editors and affiliations

  • Michael J. Ostwald
    • 1
  1. 1.University of New South WalesSydneyAustralia

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