Isidore’s Compass

A Scholium by Eutocius on Hero’s Treatise On Vaulting

in Nuncius
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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.

Isidore’s Compass

A Scholium by Eutocius on Hero’s Treatise On Vaulting

in Nuncius




Richard KrautheimerEarly Christian and Byzantine Architecture (New York: Penguin Books1981) 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 KnorrTextual Studies in Ancient and Medieval Geometry (Boston: Birkhäuser1989) p. 99.


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VitruviusDe Architectura9. 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 BD; De E apud Delphos 386 E; Quaestiones Convivales 718 E–F; Marcellus 305 EF. 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.


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Alphonse Dain“Les stratégistes byzantins,” Travaux et memoires1967 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 ConnollyGreece and Rome at War (London: Green-Hill Books1998) 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).


PhilonBelopoeica51.15–18 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 = 11 3√ (100 W).


PhilonBelopoeica52.1; 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.


PhilonBelopoeica51. 21–25. As regards the ballista Vitruvius gives ratios that correspond to Philon’s formula: De Architectura 10.11.3.


HeroBelopoeica113.1–114.3 in Marsden Technical Treatises (cit. note 22) pp. 38–41.




Julian Lowell CoolidgeA History of Geometrical Methods (Oxford: Clarendon Press1940) 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 Skrifter1938 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’Alexandrieedited 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.


SuetoniusNero41. 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 Asiatique1893 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. BallThe Domus Aurea and the Roman Architectural Revolution (Cambridge: Cambridge University Press2003) 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 OusterhoutMaster Builders of Byzantium (Princeton: Princeton University Press1999) pp. 72–74.


John WarrenGreek Mathematics and the Architects to Justinian (London: Coach Publishing1976).


ProcopiusDe aedificiis1. 1. 23–26.


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 Lettres1786 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 antichiedited 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.


HomerIliad2 v. 173; 4 v. 358; 8 v. 93.


Serafina CuomoTechnology and Culture in Greek and Roman Antiquity (Cambridge: Cambridge University Press2007) p. 206.


TacitusAnnals15. 42. 1.


PlinyNatural History35. 120.


Glanville Downey“Byzantine Architects: Their Training and Methods,” Byzantion1946–1948 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 PasqualeL’arte del costruire. Tra conoscenza e scienza (Venezia: Marsilio1996).


VitruviusDe Architectura5. 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.


AristophanesNubesv. 178; Aves v. 1003.


Paolo Galluzzi Filippo Camerota“Lo scrittoio,” in La mente di Leonardo. Nel laboratorio del genio universaleedited 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 bibliothèques1874 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 asiatiqueFévrier-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,” Isis1983 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 Mathematics10 vols. Vol. 7 (Dordrecht: Kluwer 1991) p. 64.


Carl B. BoyerA History of Mathematics (New York: Wiley1968) p. 214.


Francesca De Michelis“Parabola,” in Dizionario d’Ingegneriaedited 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 DurmDie Baukunst der Etrusken. Die Baukunst der Römer (Stuttgart: Kröner1905) pp. 301–303.


Auguste ChoisyL’art de bâtir chez les Byzantins (Paris: Sociétè anonyme de publications périodiques1883) pp. 31–43.


Ibid. p. 237.


Otto Kurz“The Date of the Tâq i Kisrâ,” Journal of the Royal Asiatic Society1941:37–41.


Tadeusz Lewicki“al-Kazwînî,” in Encyclopédie de l’Islam13 vols. Vol. 4 (Leiden/Paris: Brill-Maisonneuve & Larose 1978) pp. 898–900.


Maximilian StreckDie alte Landschaft Babylonien nach den arabischen Geographen (Leiden: Brill1901) p. 257.


ProcopiusDe bello persico1. 21 and following.


Ibid.2.8. 1–20. Procopius History of the WarsBooks 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.


ProcopiusDe bello persico2.14. 2.


ProcopiusDe bello persico2.14. 1.


Jacques Heyman“Poleni’s Problem,” Proceedings of the Institution of Civil Engineers1988 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 Bulletin1986 68.1:24–34.


Friedrick Rakob“Römische Kuppelbauten in Baiae,” Mitteilungen des Deutschen Archäologischen Instituts Römische Abteilung1988 95:257–301 particularly Plates 13–14.


Robert HookeA Description of Helioscopes and some other Instruments (London: J. Martyn for the Royal Society1675) 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 GalileiDiscorsi 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: Elsevirii1638) 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 Transactions1698 19:637–652. Di Pasquale L’arte (cit. note 79) pp. 267–268.


Edoardo BenvenutoLa scienza delle costruzioni e il suo sviluppo storico (Firenze: Sansoni, 1981; reprint Roma: Edizioni di Storia e Letteratura2006) 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 StirlingLineae Tertii Ordinis Neutonianae sive Illustratio Tractatus D. Neutoni De Enumeratione Linearum Tertii Ordinis. Cui subjugitur Solutio Trium Problematum (Oxoniae: Whistler1717) 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 PoleniMemorie istoriche della gran cupola del Tempio Vaticano e de’ danni di essa e de’ ristoramenti loro divise in libri cinque (Padova: Stamperia del Seminario1748) 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 construcciónedited by Daniel Giralt-Miracle (Barcelona: Lunwerg 2002) pp. 96–103.


  • View in gallery
    Figure 1

    Vatican Library, Vatican City, Ottob. lat. 1850, f. 37v, detail: Menaechmus’ solution for doubling the cube by means of two parabolas.

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    Figure 2

    Bibliothèque Nationale de France, Paris, Persan 169, f. 148, detail: Aboûl Wafâ’s construction of a parabola to make a burning mirror.

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    Figure 3

    Left – Diocles: using pointwise construction to draw a parabola (from Rashed). Right – The device probably used by Isidore to draw a parabola (by the author).

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    Figure 4

    Left – Ibn Sahl’s geometrical device to draw a parabola (by the author). Right – Geometrical device to draw a parabola, by Francesca De Michelis (from Perucca, 1954).

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    Figure 5

    Iraq, Ctesiphon, the Arch of Kosrow (from Reuther, 1929).

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    Figure 6

    James Stirling, “Methodus disponendo quotcumque Sphaeras in Fornicem. Et inde Demonstratur Proprietas praecipua Curvae Catenariae” (from Stirling, 1717).


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