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In the history of architecture, the construction of vaults was one of Rome’s supreme achievements, yet paradoxically it is not mentioned in any of the existing literary sources. Hero of Alexandria wrote a treatise entitled On Vaulting, which unfortunately is lost. A commentary on it was written by Isidore of Miletus, who lived at the time of the construction of the Hagia Sophia in Constantinople. We know this only from a brief remark in Commentary on the Sphere and Cylinder of Archimedes, written by Eutocius of Ascalon; furthermore the scholium refers to the drawing of a parabola. It was at about this same time that the parabola became very important in the East, as can be seen in the construction of Kosrow’s Arch at Ctesiphon. The present study examines Eutocius’ remark and the context in which it is to be found in his Commentary, and takes a look at the primacy of the parabola in the static behaviour of domes.
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Richard Krautheimer, Early Christian and Byzantine Architecture (New York: Penguin Books, 1981), pp. 215, 220. More recently, Lucio Russo, La rivoluzione dimenticata. Il pensiero scientifico greco e la scienza moderna (Milano: Feltrinelli, 1996), pp. 285–286.
Wilbur Richard Knorr, Textual Studies in Ancient and Medieval Geometry (Boston: Birkhäuser, 1989), p. 99.
Alan Cameron, “Isidore of Miletus and Hypatia. On the Editing of Mathematical Texts,” Greek, Roman and Byzantine Studies, 1990, 30:103–127, p. 107. Moreover Knorr, Textual Studies (cit. note 5), pp. 98–100.
Vitruvius, De Architectura, 9. Praef. 13. Vitruve, De l’Architecture, Livre IX, edited by Jean Soubiran (Paris: Les Belles Lettres, 1969), pp. 57–59: in note 36 the editor compares texts by Vitruvius and Plutarch. Plutarch, De Genio Socratis, 579 B–D; De E apud Delphos, 386 E; Quaestiones Convivales, 718 E–F; Marcellus, 305 E–F. Aristoula Georgiadou, “The Corruption of Geometry and the Problem of Two Mean Proportionals,” in Plutarco e le scienze, edited by Italo Gallo (Genova: Sagep, 1992), pp. 147–164.
Liba Taub, “‘Eratosthenes sends greetings to King Ptolemy’: Reading the Contents of a ‘Mathematical’ Letter,” in Mathematics Celestial and Terrestrial. Festschrift für Menso Folkerts zum 65. Geburgstag, edited by Joseph W. Dauben, Stefan Kirschner, Andreas Kühne, Paul Kunitzsch, Richard Lorch (Halle: Deutsche Akademie der Naturforscher Leopoldina, 2008), pp. 285–302.
Alphonse Dain, “Les stratégistes byzantins,” Travaux et memoires, 1967, 2:317–392, pp. 323–324. Giovanni Di Pasquale, Tecnologia e meccanica. Trasmissione dei saperi tecnici dall’età ellenistica al mondo romano (Firenze: Olschki, 2004), pp. 89–124.
Peter Connolly, Greece and Rome at War (London: Green-Hill Books, 1998), p. 282. Flavio Russo, L’artiglieria delle legioni romane (Roma: Libreria dello Stato, 2004). Rubén Sáez Abad, Artillería y poliorcética en el mundo grecorromano (Madrid: Polifemo, 2005).
Philon, Belopoeica, 51.15–18, in Eric William Marsden, Greek and Roman Artillery. Technical Treatises (Oxford: Clarendon Press, 1971), pp. 108–109. Id., Greek and Roman Artillery. Historical Development (Oxford: Clarendon Press, 1969), pp. 24–25: D = 1,1 3√ (100 W).
Philon, Belopoeica, 52.1; as regards the geometrical construction for doubling the cube, the writer refers to his first book, now lost; however, the construction has survived thanks to Eutocius. Serafina Cuomo, “L’età classica ed ellenistica,” in La matematica, edited by Claudio Bartocci, Piergiorgio Odifreddi, 4 vols., Vol. 1 (Torino: Einaudi, 2007), pp. 37–63: especially pp. 43–47. Gian Arturo Ferrari, “Meccanica ‘allargata’,” in La scienza ellenistica, edited by Gabriele Giannantoni, Mario Vegetti (Napoli: Bibliopolis, 1985), pp. 225–296.
Philon, Belopoeica, 51. 21–25. As regards the ballista, Vitruvius gives ratios that correspond to Philon’s formula: De Architectura, 10.11.3.
Hero, Belopoeica, 113.1–114.3, in Marsden, Technical Treatises (cit. note 22), pp. 38–41.
Julian Lowell Coolidge, A History of Geometrical Methods (Oxford: Clarendon Press, 1940), p. 119; the author suggests that the origins of algebraic geometry are to be found in Menaechmus’ studies. Knorr, Textual Studies (cit. note 5), p. 117.
Pappus, Vol. 3 (cit. note 19), p. 1022. 14–15. Cuomo, Pappus (cit. note 14), pp. 93–94.
Otto E. Neugebauer, “Uber eine Methode zur Distanzbestimmung Alexandria – Rom bei Heron,” Kongelige Danske Videnskabernes Selskabs Skrifter, 1938, 26.2:21–24.
Dimitris Raïos, “La date d’Héron d’Alexandrie: témoignages internes et cadre historico-culturel,” in Autour de la Dioptre d’Héron d’Alexandrie, edited by Gilbert Argoud, Jean-Yves Guillaumin (Saint-Étienne: Université de Saint-Étienne, 2000), pp. 19–36; particularly p. 36, where the author dates the publication of Automata to before 67AD.
Micheline Decorps-Foulquier, “Remarques liminaires sur le texte de la Dioptre de Héron d’Alexandrie et ses sources,” in Autour de la Dioptre d’Héron (cit. note 35), pp. 37–43.
Pierre Souffrin, “Remarques sur la datation de la Dioptre d’Héron par l’éclipse de lune de 62,” in Autour de la Dioptre d’Héron (cit. note 35), pp. 13–17.
Suetonius, Nero, 41. 2. Paul Keyser, “Suetonius Nero 41.2 and the date of Heron Mechanicus of Alexandria,” Classical Philology, 1988, 83:218–220.
Bernard Carra de Vaux, “Héron d’Alexandrie, Les Mécaniques ou l’élévateur des corps lourds,” Journal Asiatique, 1893, 1:386–472; 1893, 2:152–192, 227–269, 461–514; new edition by Donald R. Hill, Aage Gerhardt Drachmann (Paris: Les Belles Lettres, 1988). Aage Gerhardt Drachmann, The Mechanical Technology of Greek and Roman Antiquity (Copenhagen and Madison: The University of Wisconsin Press and Hafner Press, 1963).
Larry F. Ball, The Domus Aurea and the Roman Architectural Revolution (Cambridge: Cambridge University Press, 2003), pp. 207–218. Lynne C. Lancaster, Concrete Vaulted Construction in Imperial Rome: Innovation in Context (Cambridge: Cambridge University Press, 2005), pp. 42–43. Cinzia Conti, Giangiacomo Martines, Anna Sinopoli, “Constructions Techniques of Roman Vaults: Opus Caementicium and the Octagonal Dome of the Domus Aurea,” in Proceedings of the Third International Congress on Construction History, edited by Karl-Eugen Kurrer, Werner Lorenz, Volker Wetzk, 2 vols., Vol. 1 (Cottbus: Brandenburg University of Technology, 2009), pp. 401–408. Cinzia Conti, “Domus Aurea, Octagonal Hall: Analysis of Materials,” in Mechanics and Architecture between Epistéme and Téchne, in Commemoration of Edoardo Benvenuto, edited by Anna Sinopoli (Roma: Edizioni di Storia e Letteratura, 2010), pp. 94–96.
Ibid., pp. 46.21–48.10. The unit of measurement for the seat corresponds to one foot: ἕκαστος γὰρ ποὺς ἕνα ἄνδρα χωρεῖ.
Robert Ousterhout, Master Builders of Byzantium (Princeton: Princeton University Press, 1999), pp. 72–74.
John Warren, Greek Mathematics and the Architects to Justinian (London: Coach Publishing, 1976).
The first edition by Louis Dupuy, “Fragment d’un ouvrage grec d’Anthémius sur les paradoxes de mécanique. Revu et corrigé sur quatre manuscrits, avec une traduction françoise, des notes critiques et des observations, et les variantes tirées d’un manuscrit du Vatican,” Histoire de l’Academie des Inscriptions et Belles Lettres, 1786, 42:392–451. George Leonard Huxley, “Anthemius of Tralles. A Study in Later Greek Geometry,” Greek Roman and Byzantine Monographs, 1959, 1.
Filippo Coarelli, “Il faro di Alessandria,” in Eureka. Il genio degli antichi, edited by Eugenio Lo Sardo (Napoli: Electa, 2005), pp. 85–90; Paolo Vitti, Ottavio Voza, “Note a margine di un disegno sul faro di Alessandria,” in ibid., pp. 84 and 91.
Serafina Cuomo, Technology and Culture in Greek and Roman Antiquity (Cambridge: Cambridge University Press, 2007), p. 206.
Glanville Downey, “Byzantine Architects: Their Training and Methods,” Byzantion, 1946–1948, 18:99–118, quoted from p. 111 and p. 109. The Decree of 6 July 344 is in: Theodosiani libri XVI cum constitutionibus Sirmondianis, edited by Theodor Mommsen, Paul Martin Meyer, 2 vols., Vol. 1.2 (Berlin: Weidmann, 1905), p. 747: XIII. 4. 3. The Theodisian Code and Novels and the Sirmondian Constitutions, edited by Clyde Pharr (New York: Greenwodd, 1952), p. 391. Codex Justinianus, edited by Paul Krueger, 3 vols., Vol. 2 (Berlin: Weidmann, 1929), p. 425: X. 66. 2.
Pappus (cit. note 19), Vol. 3, p. 1028. 4–5. Cuomo, Pappus (cit. note 14), pp. 104–109.
Salvatore Di Pasquale, L’arte del costruire. Tra conoscenza e scienza, (Venezia: Marsilio, 1996).
Vitruvius, De Architectura, 5. 10.3: concamerationes; 7. 2.2: camerarum dispositiones; 7. 3.1: camerarum ratio; 7. 10.2: camerae curvatura; 8. 2.4: camera. Ettore Bosazzi, Il ‘De Architectura’ di Vitruvius. Studi sulla lingua (Trieste: Editreg, 2000), p. 24.
Hero (cit. note 34), p. 140. 3.
Paolo Galluzzi, Filippo Camerota, “Lo scrittoio,” in La mente di Leonardo. Nel laboratorio del genio universale, edited by Paolo Galluzzi (Firenze: Giunti, 2006), pp. 146–149.
M. François Woepcke, “Trois traités arabes sur le compas parfait,” Notices et extraits des manuscrits de la Bibliothèque Impériale et autres bibliothèques, 1874, 22.1:1–175; here the codex is given its previous numbering: Arabe 1076. Woepcke’s work was published by Jules Mohl, with an introduction by M. de Slane. The Journal asiatique, 1864:17–24, published a biography of François Woepcke, who died at a young age, and a list of his works.
Ibid., p. 68.
Ibid., pp. 16–18, 111.
Ibid., p. 87.
François Woepcke, “Analyse et extrait d’un recueil de constructions géometriques par Aboûl Wafâ,” Journal asiatique, Février-Mars 1855:218–256; Avril 1855:309–359.
Ibid., pp. 325–327. Otto Neugebauer, Roshdi Rashed, “Sur une construction du miroir parabolique par Abû al-Wafâ’ al-Bûzjânî,” Arabic Sciences and Philosophy, 1999, 9.2:261–277. In Figure 2, the chords of the circle, which come out from the vertex, have been transferred onto corresponding parallels one-to-one, each chord with one end on the axis while the other ends, taken together, draw a parabola.
Ibid., pp. 53–57. In Figure 3 (left), the vertex of the right angle is the focus of the parabola. Let us draw a segment of any kind inside the triangle and parallel to the directrix; now, with the compass centre on the focus and the compass opening equal to the segment, we mark a point P on the segment itself: this point forms part of the parabola to be constructed, because it is equidistant from the focus and directrix.
Wilbur Knorr, “The Geometry of Burning-Mirrors in Antiquity,” Isis, 1983, 74:53–73, p. 56. Diocles’ Proposition 10 corresponds to Menaechmus’ second solution for doubling the cube, through the intersection of the two parabolas: Rashed, Les catoptriciens (cit. note 63), pp. 79–82; however, Diocles’ solution is not mentioned at all by Eutocius: Knorr, Textual Studies (cit., note 5), pp. 94, 100, 115.
A.B. Ivanov, “Parabola,” in Encyclopaedia of Mathematics, 10 vols., Vol. 7 (Dordrecht: Kluwer, 1991), p. 64.
Carl B. Boyer, A History of Mathematics (New York: Wiley, 1968), p. 214.
Francesca De Michelis, “Parabola,” in Dizionario d’Ingegneria, edited by Eligio Perucca, vols. 5, Vol. 4 (Torino: Utet, 1954), s.v.
Joannes Malala, “Chronographia,” in Patrologiae cursus (cit. note 59), Vol. 97 (1860), columns 661–662. Jeffreys, Jeffreys, Scott, The Chronicle (cit. note 59), p. 264. 47; the dates of the events referred to later in this paper are based on this publication.
Josef Durm, Die Baukunst der Etrusken. Die Baukunst der Römer (Stuttgart: Kröner, 1905), pp. 301–303.
Auguste Choisy, L’art de bâtir chez les Byzantins (Paris: Sociétè anonyme de publications périodiques, 1883), pp. 31–43.
Ibid., p. 237.
Otto Kurz, “The Date of the Tâq i Kisrâ,” Journal of the Royal Asiatic Society, 1941:37–41.
Tadeusz Lewicki, “al-Kazwînî,” in Encyclopédie de l’Islam, 13 vols., Vol. 4, (Leiden/Paris: Brill-Maisonneuve & Larose, 1978), pp. 898–900.
Maximilian Streck, Die alte Landschaft Babylonien nach den arabischen Geographen (Leiden: Brill, 1901), p. 257.
Ibid., 2.8. 1–20. Procopius, History of the Wars, Books 1 and 2, edited by H.B. Dewing (London/Cambridge, Mass.: Loeb, 1914), pp. 325–331. Marie Louise Chaumont, “Antioch,” in Encyclopaedia Iranica, Vol. 2 (cit. note 125), pp. 119–125: particularly pp. 123–124.
Jacques Heyman, “Poleni’s Problem,” Proceedings of the Institution of Civil Engineers, 1988, 84.1:737–759. In a hemispheric dome, the theory of curved membranes identifies the limit of the zone of tension – in the lower part – 38° azimuth angle from the springing level; while in the upper part the parallels are subject only to compression; Odone Belluzzi, Scienza delle costruzioni, 4 vols., Vol. 3 (Bologna: Zanichelli, 1947), pp. 267–269.
Robert Mark, Paul Hutchinson, “On the Structure of the Roman Pantheon,” The Art Bulletin, 1986, 68.1:24–34.
Friedrick Rakob, “Römische Kuppelbauten in Baiae,” Mitteilungen des Deutschen Archäologischen Instituts, Römische Abteilung, 1988, 95:257–301, particularly Plates 13–14.
Robert Hooke, A Description of Helioscopes and some other Instruments (London: J. Martyn for the Royal Society, 1675), p. 31; the next page contains the famous “Hooke’s Law”: “ut pondus sic tensio.” I consulted the copy in the Bibliothèque Nationale del France, Paris, in the section Tolbiac-Rez-de-Jardin, Salle Y.
Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla meccanica & i movimenti locali. Con una Appendice del centro di gravità d’alcuni solidi (Leida: Elsevirii, 1638), pp. 282–288, Propos. XIV, particularly p. 284. Le Opere di Galileo Galilei, edited by Antonio Favaro, 21 vols., Vol. 8 (Firenze: Barbera, 1898), pp. 309–310. On the difference between the parabola and the catenary: Renato Sparacio, La scienza e i tempi del costruire (Torino: Utet, 1999), pp. 155–158.
David Gregory, “Catenaria,” Philosophical Transactions, 1698, 19:637–652. Di Pasquale, L’arte (cit. note 79), pp. 267–268.
Edoardo Benvenuto, La scienza delle costruzioni e il suo sviluppo storico (Firenze: Sansoni, 1981; reprint Roma: Edizioni di Storia e Letteratura, 2006), pp. 182–200. Jacques Heyman, The Masonry Arch (Chichester: Horwood, 1982). Alessandro Becchi, Federico Foce, Degli archi e delle volte. Arte del costruire tra meccanica e stereotomia (Venezia: Marsilio, 2002).
James Stirling, Lineae Tertii Ordinis Neutonianae sive Illustratio Tractatus D. Neutoni De Enumeratione Linearum Tertii Ordinis. Cui subjugitur, Solutio Trium Problematum (Oxoniae: Whistler, 1717), pp. 11–14. I consulted the copy in the Biblioteca unificata della Scienza e della Tecnica dell’Università degli Studi di Pavia, Sezione di Matematica.
Giovanni Poleni, Memorie istoriche della gran cupola del Tempio Vaticano e de’ danni di essa, e de’ ristoramenti loro, divise in libri cinque (Padova: Stamperia del Seminario, 1748), Tav. D, Fig. XI. The sequence Gregory, Stirling, Poleni was noted by Di Pasquale, L’arte (cit. note 79), p. 268.
Gottfried Wilhelm von Leibniz, “De linea in quam flexile se pondere proprio curvat, eiusque usu insigni ad inveniendas quotcumque medias proportionales et Logarithmos,” Acta Eruditorum (Lipsiae: Gunther), Mensis Junii 1691:277–281. I consulted the copy in the Vatican Library.
Ibid., pp. 1119–1123.
Josep Gómez-Serrano, “Arcos catenarios,” in Gaudí. La búsqueda de la forma. Espacio, geometría, estructura y construcción, edited by Daniel Giralt-Miracle (Barcelona: Lunwerg, 2002), pp. 96–103.
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In the history of architecture, the construction of vaults was one of Rome’s supreme achievements, yet paradoxically it is not mentioned in any of the existing literary sources. Hero of Alexandria wrote a treatise entitled On Vaulting, which unfortunately is lost. A commentary on it was written by Isidore of Miletus, who lived at the time of the construction of the Hagia Sophia in Constantinople. We know this only from a brief remark in Commentary on the Sphere and Cylinder of Archimedes, written by Eutocius of Ascalon; furthermore the scholium refers to the drawing of a parabola. It was at about this same time that the parabola became very important in the East, as can be seen in the construction of Kosrow’s Arch at Ctesiphon. The present study examines Eutocius’ remark and the context in which it is to be found in his Commentary, and takes a look at the primacy of the parabola in the static behaviour of domes.
All Time | Past Year | Past 30 Days | |
---|---|---|---|
Abstract Views | 388 | 56 | 5 |
Full Text Views | 230 | 1 | 0 |
PDF Views & Downloads | 22 | 1 | 0 |