Hero of Alexandria and Mordekhai Komtino: The Encounter between Mathematics in Hebrew and the Greek Metrological Corpus in Fifteenth-Century Constantinople Tony Lévy, Bernard Vitrac
To cite this version:
Tony Lévy, Bernard Vitrac. Hero of Alexandria and Mordekhai Komtino: The Encounter between Mathematics in Hebrew and the Greek Metrological Corpus in Fifteenth-Century Constantinople. Aleph, Indiana University Press, 2018. hal-03328208
HAL Id: hal-03328208 https://hal.archives-ouvertes.fr/hal-03328208 Submitted on 29 Aug 2021
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tony Lévy and Bernard Vitrac
Hero of Alexandria and Mordekhai Komtino: The Encounter between Mathematics in Hebrew and the Greek Metrological Corpus in Fifteenth- Century Constantinople
ABSTRACT Mordekhai ben Eliezer Komtino (1402–1482) was a well-known scholar in the Judeo-Byzantine world—philosopher, exegete, mathematician, and astronomer. Among his works was Sefer ha-Ḥeshbon we-ha-middot. This paper deals with the geometrical part of that work. Although Euclid is sometimes quoted by the author, Komtino’s geometry is not Euclidean. It comes from the field of metrological geometry (“practical geometry”). In addition to the famous Hebrew geometry composed by Abraham bar Ḥiyya in the twelfth century, Ḥibbur ha-meshiḥah we-ha-tishboret, Komtino also had before him a Greek manuscript containing the unique extant copy of Hero of Alexandria’s Metrica (first or second century) and other metrological writings attributed to him.
Mordekhai Komtino’s Place in the History of the Hebrew Scientific Culture: The State of the Question Scientific literature in Hebrew, and notably its mathematical branch, originated
Tony Lévy is at the CNRS, UMR 7219 (SPHERE), email: [email protected]. Bernard Vitrac is at the CNRS UMR 8210 (AnHiMA), email: [email protected].
© Aleph 18.1 (2018) pp. 181-262 181 Hero of Alexandria and Mordekhai Komtino at the start of the twelfth century1 and continued through the sixteenth century, in very different places, eras, and cultural contexts.2 In this paper we will present a rather late and highly interesting link in this tradition: Rabbi Mordekhai Komtino in fifteenth-century Constantinople.
Komtino in his Milieu Mordekhai ben Eliezer Komtino, also spelled Komtiano or Komatiano3 (1402– 1482), is one link in the line of Jewish polymaths who, in addition to their commentaries on canonical texts, also wrote on the sciences;4 in his case, astronomy and mathematics. We should also underscore the frequent references
1 Tony Lévy, “The Establishment of the Mathematical Bookshelf of the Medieval Hebrew Scholar,” Science in Context 10(3) (1997): 431–451. 2 Tony Lévy, “The Hebrew Mathematics Culture (Twelfth-Sixteenth Centuries),” in Gad Freudenthal, ed., Science in Medieval Jewish Cultures (Cambridge: Cambridge University Press, 2011), pp. 155–171, on p. 160: “Mathematics in the Judeo-Byzantine Scholarly World (Fifteenth-Sixteenth Centuries).” 3 As summarized by Jean-Christophe Attias, Le commentaire biblique. Mordekhai Komtino ou l’herméneutique du dialogue (Paris: Le Cerf, 1991), “The exact pronunciation of seems to have divided researchers. Its Hebrew orthography is (כומטינו) Komtino’s name certainly fluctuating and seems to authorize several readings” (p. 20); see also nn. 3–12. see Steven ;(כומאטייאנו) Langermann, following Bowman, opts for the reading Komatiano Bowman, The Jews of Byzantium 1204–1453 (Tuscaloosa: University of Alabama Press, 1985) p. 149, n.68; Y. Tzvi Langermann, “Science in the Jewish Communities of the Byzantine Cultural Orbit. New Perspectives,” in Freudenthal, ed., Science in Medieval Jewish Cultures, pp. 438–453, on p. 438, n.1. Other Hebrew representations are attested We note, however, that the oldest manuscript of the .(כומיטינו/חומטייאנו/כומטייאנו) mathematical text analyzed here (see below, the colophon of the New York manuscript JTS 2639) was copied in the author’s lifetime, in 1478; there the copyist presents the author may God protect and give him life, the ,(כומטיינו) by the scribe as “Mordechai Komtiano son of Eliezer Komtiano.” We do not feel adequate to decide the question, which depends on more than paleography. We have chosen to go here with Komtino. 4 For a first overview of the scientific activity of the Judeo-Byzantine communities between the eight and sixteenth centuries, see Langermann, “Science in the Jewish Communities of the Byzantine Cultural Orbit.” With regard to Komtino, Langermann writes (p. 439): “For the most part, his writings remain unstudied; if they ever are closely studied, surely our picture of Byzantine Jewish culture will benefit.”
182 Tony Lévy and Bernard Vitrac to the sciences in his exegetical works, including his commentary on the Pentateuch,5 his commentary on works by Abraham Ibn Ezra,6 his glosses on Maimonides’ Treatise on Logic, known in Hebrew as Millot ha-higgayon,7 and his commentary on the Guide of the Perplexed (completed in 1480).8 The biographical data about Komtino are sparse and fragmentary.9 He lived in a century that was marked by a crucial event: the Ottoman expansion and conquest of Constantinople in 1453, which had a powerful effect on the Judeo- Byzantine communities.10 On the cultural and thus on the scientific plane, these communities’ encounter with the stream of exiles from the Iberian Peninsula (from the end of the fourteenth and continuing into the sixteenth century), with their influential traditions, was a major turning point. Finally, we should underscore Komtino’s special relationship with the Karaites of Constantinople (Jewish sectarians who deny the authority of the rabbinic Oral Law); several of its members were among his students. This circumstance no doubt encouraged Komtino’s scholarly work, as evidenced by the fact that several of his works are dedicated to his Karaite students.11
5 Attias, Le commentaire biblique, pp. 72–73. 6 Sefer Yesod mora’ (Book of the Foundation of Piety), Sefer ha-Shem (The Book of the Name), and Sefer ha-Eḥad (The Book of the One). For the first of these commentaries by Komtino, see Dov Schwartz, Perush qadmon le-sefer Yesod moraʾ (An Early Commentary on Yesod mora) (Ramat Gan: Bar-Ilan University, 2010). 7 This text, the only work by Komtino published before the twenty-first century, is a pendant to Maimonides’ text, along with the commentary by Moses Mendelssohn: Millot ha-higgayon, ed. David Slucki (Warsaw: Baumritter, 1865). See Charles H. Manekin, “Logic in Jewish Medieval Culture,” in Freudenthal, ed., Science in Medieval Jewish Cultures, p. 129. 8 Dov Schwartz “Understanding in Context: Rabbi Mordechai Komtino’s Commentary on the Guide of the Perplexed,” Pe‘amim 133–134 (2013): 127–183 (Heb). 9 Attias, Le commentaire biblique, pp. 19–22. 10 Joseph Hacker, “Ottoman Policy toward the Jews and Jewish Attitudes toward the Ottomans during the Fifteenth Century,” in Benjamin Braude and Bernard Lewis, eds., Christians and Jews in the Ottoman Empire, vol. 1 (New York: Holmes & Meier, 1982), pp. 117–126; idem, “The Sürgün System and Jewish Society in the Ottoman Empire during the Fifteenth to the Seventeenth Centuries,” in Aron Rodrigue, ed., Ottoman and Turkish Jewry (Bloomington: Indiana University Press, 1992), pp. 1–66. 11 Attias, Le commentaire biblique, pp. 15–16, 30–31, 41–44.
183 Hero of Alexandria and Mordekhai Komtino
הקושטנדיני—”Komtino represents himself as “a Greek of Constantinople His command of Greek is attested in his commentaries on the Pentateuch .היוני and on Maimonides and in his celestial catalogue, which provides the names of stars in both Greek and Hebrew.12 Our study of his geometrical treatise confirms his fluency in Greek and his concern to convey its nuances. What scientific heritage could Komtino have had in his day? Many older manuscripts were being copied in Constantinople/Istanbul; translations into Hebrew were being executed from Arabic, Greek, and even directly from Persian.13 Alongside a number of scientific encyclopedias compiled in Hebrew there were also original writings, with a three-fold heritage: the works of Abraham Bar Ḥiyya and Abraham Ibn Ezra; the translations of Arabic works that issued from Provence; and treatises such as those by Gersonides and Immanuel Bonfils. To this we must add works in Arabic and Greek that were not available in Hebrew.
Mordekhai Komtino’s Scientific Writings The pioneering labors of Ḥayyim Jonah Gurland (1843–1890), a Russian rabbi and orientalist, produced the first overview of Komtino’s writings, which he accessed in the manuscript collection of the Imperial Library in St. Petersburg.14
12 Ibid., pp. 31–33. According to Attias, the glosses in Arabic do not necessarily reflect a real knowledge of the language. Such knowledge does seem to be attested for one of his most lauded pupils, Eli Mizrachi (1435–1526), the author, among other works, of a substantial commentary on the Almagest. In the latter Langermann has identified many exact and accurate references to both the Greek and Arabic versions of the Almagest; see Langermann, “Science in the Jewish Communities,” p. 447. 13 This is the case, for now unique, of a medical formulary by Zayn al-Dīn al-Jurjānī (d. 1136/7); see ibid, p. 443. 14 Gurland published the results of his studies in a monograph in Russian: Novye Matierialy dlia istorii Ievrieïskoï Literatury XV Stolietia. M. Koumatiano (New materials on the history of Hebrew literature in the 15th century by M. Koumatiano) (St Petersburg: Imperial Academy of Sciences, 1866). It was accompanied by a substantial appendix of excerpts from the Hebrew manuscripts consulted by Gurland. The appendix was published separately, under the Hebrew title Ginzei Yisra’el with an additional title page in German: Neue Denkmäler der jüdischen Literatur in St. Petersburg, vol. 3 (Lyck, 1865). At Gurland’s own request, the Russian-language monograph was translated into Hebrew and revised by Judah Levi Lewick. The Hebrew text was published after Gurland’s death in
184 Tony Lévy and Bernard Vitrac
Komtino produced a number of treatises on the mathematical sciences, especially astronomy.15
In it he 16.(ספר פירוש לוחות פרס) A commentary on the Persian Tables • ,(מפרשי יון החדשים) ”mentions a number of “recent Greek commentators whom he censures for his 17,(אריירו) such as the monk Isaac Argyre groundless criticism of the quality of these tables.
Three other short extant works deal with the construction of astronomical instruments:
the literary anthology Talpiot (Berditchev, 1895). The eleventh section of this anthology (of its thirteen), entitled “The History of Famous Men” includes a 34-page chapter (paginated separately), “The History of Rabbi Mordechai Komtino,” divided into three parts: “The History of Mordechai Komtino and the Scholars of his Time” (pp. 3–7); “The Breadth, scope, and Power of M. Komtino” (pp. 7–14); and “Komtino’s Works” (pp. 14–33), with 11 titles. We learn there (p. 8) that Komtino was educated in the school of great rabbis, but “only one of them is known to us by name, R. Hanokh Saporta, one of the great rabbis of Catalonia.” Gurland also published a brief description of the scientific manuscripts in the Firkowitz collection, Kurze Beschreibung der mathematischen, astronomischen und astrologischen hebräischen Handschriften der Firkowitsch’schen Sammlung … (St. Petersburg, 1866). 15 Attias, Le commentaire biblique, pp. 75–79: “Le cas de l’astronomie.” Komtino’s students, such as the Rabbanite Eli Mizrahi (who became chief rabbi of Constantinople) and the Karaite Caleb Afendopulo (who became a leader of his community) went on to produce major works in astronomy. 16 A Hebrew version of these tables was produced by Solomon b. Eli of Salonica (fl. 1374–1386) from the Greek recension by George Chrysococces (before 1347); see Bernard R. Goldstein, “The Survival of Arabic Astronomy in Hebrew,” Journal for the History of Arabic Science 3 (1979): 31–39, on p. 36–37; idem, “Science in Medieval Jewish Communities,” in Gilbert Dahan, ed., Les juifs au regard de l’histoire. Mélanges en l’honneur de Bernhard Blumenkranz (Paris: Picard, 1985), pp. 235–247, on pp. 240–241. On the sources of this Greek version, see: David Pingree, “Gregory Chionades and Palaeologan Astronomy,” Dumbarton Oaks Papers 18 (1964): 135–160, on pp. 141–145; Anne Tihon, “Les tables astronomiques persanes à Constantinople dans la première moitié du XIVe siècle,” Byzantion. Revue Internationale des études byzantines 57 (1987): 471–487. 17 Excerpts were published in Gurland, Ginzei Yisra’el, p. 2. For a brief summary of Komtino’s text, see Gurland/Lewick, Talpiot, pp. 17–18.
185 Hero of Alexandria and Mordekhai Komtino
18(תקון כלי הנחשת) On the Construction of the Astrolabe • ,This instrument 19.(תקון הכלי הצפיחה) On the Construction of the Ṣafiḥah • invented in the eleventh century by the Andalusian astronomer al-Zarqālluh, consists of a circular plate (Arabic ṣafiḥa = plate) that could be used to track the positions of the planets (an equatorium). תקון כלי) On the Construction of the Instrument [to determine] the Hours • with instructions for building two different types of sundials.20 ,(השעות
In addition to the treatise on arithmetic and geometry that is the subject of the present study, we should also mention Komtino’s notes on the Almagest, which include occasional references to Euclid’s Elements.21 These notes survive on folios 29a–33a of a manuscript in the Russian State Library in Moscow.22 The manuscript itself is a medley of commentaries or excerpts from commentaries, most of them on three of the favorite mathematical works of Paleologue Byzantium: Nicomachus’ Introduction to Arithmetic, Euclid’s Elements, and
18 Gurland/Lewick, Talpiot, pp. 18–20. 19 Ibid., pp. 20–21. Komtino could have had access to the Hebrew text of al-Zarqālluh’s Arabic work, translated, no doubt, by Jacob ben Machir; see Lévy,“The Establishment of the Mathematical Bookshelf,” p. 442 n. 30. 20 Gurland/Lewick, Talpiot, p. 21. There is a brief description of this last text in Bernard R. Goldstein, “Descriptions of Astronomical Instruments in Hebrew,” in David King and George Saliba , eds., From Deferent to Equant (New York: New York Academy of Sciences, 1987), pp. 105–141, on p. 123. 21 These references to Euclid’s Elements were noted by Moritz Steinschneider, under the title “Glossen zu Euklid” in his Mathematik bei den Juden, vol. 1 (Hildesheim: Olms, 1964), p. 198. 22 Moscow, MS RSL Günzburg 340 (F 47 750), 16th c., 119 fols. Contents: fols. 2a–18b, (anon.), commentary on Nicomachus’ Introduction to Arithmetic; fols. 19a–23b, Abraham Yerushalmi, commentary on Nicomachus’ Introduction to Arithmetic; fols. 24a–25b, Abraham Yerushalmi, commentary on al-Farghānī’s Astronomy; fols. 26a–b, note on “two lines that never meet (the asymptotic property of the hyperbola)”; fols. 27a–28b, notes on the Almagest; fols. 29a–33a, notes on the Almagest; fols. 34a–44a, extracts from Ibn al-Haytham’s Muṣādarāt (commentaries on the premises of Books V, VI, VII, X, and XI Euclid’s Elements; text translated into en Hebrew by Moses Ibn Tibbon, dated Av 5030 [= August 1270]). The text breaks off in the middle of the commentary on the arithmetic premises; fols. 45a–119b, (anon.), commentary on the Almagest.
186 Tony Lévy and Bernard Vitrac
Ptolemy’s Almagest. The notes on the last of these that are ascribed to Komtino deal with books II (§§11, 13), III (§10), IV (§6), V (§§5, 19), VI (§7), and XIII (§§4, 6). In his comments on Almagest IV.6 (fol. 30b) and XIII.4 (fol. 33a), Komtino mentions Euclid’s Elements, Propositions III.36 (numbered 35) and V.8, respectively. In a discussion of eccentrics (Almagest V.5, fol. 31b), the author of the note, probably Komtino, quotes several lines by “Pappus the geometer” ,evidently a reference to Pappus’ commentary on the Almagest ,(פפו התשבורתי) of which books V and VI have survived in Greek.23 Still with regard to the transmission history of the Elements, we should note that the section of Komtino’s book that deals with arithmetic (not studied here) presents an arithmetical formulation of Elements II.1–10.24
Komtino’s Mathematical Treatise: The State of the Question Steinschneider listed the treatise as an untitled work on mathematics, in two parts (see reference in n. 21). In his introduction, Komtino enumerates its objectives and structure: a first part on arithmetic and a second part on geometry. We know of ten extant manuscripts, more or less complete:25
1. Berlin, Staatsbibliothek Or. Qu. 308; cat. Steinschneider No.49 (IMHM, F 1734), 15th–16th c., 189 pp., Byzantine script: complete, but with many diagrams missing. 2. London, BL, Add. 27107 Alm 213 (F 5782); cat. Margoliouth 1016, vol. III, p. 342), 16th c., fols. 44a–80b: lacking almost all the diagrams, even though space was left for them.
23 This quotation from Pappus was noted by Y. Tzvi Langermann, “Medieval Hebrew Texts on the Quadrature of the Lune,” Historia Mathematica 23 (1996): 31–53, on p. 52. Langermann underlines the possibility of translations into Hebrew made directly from the Greek sources and cautiously suggests, with regard to the passage from Pappus, that “its author may easily be presumed to have been fluent in Greek” (“Science in the Jewish Communities,” p. 449). Langermann also notes another reference to “Pappus the geometer” in the margin of a copy of the Hebrew version of the Almagest. 24 See, for example, Saint Petersburg, Firkowitz, MS Evr. I 320b, fols. 7b–8b. 25 Steinschneider catalogued five manuscripts, including Berlin, London, Paris, and St Petersburg 343, as well as New York 2639, which at the time was in the collection of A. Lehren in Amsterdam.
187 Hero of Alexandria and Mordekhai Komtino
3. New York, JTS 2632 (= ENA 1574; F 28 885), 19th–20th c., 46 pp. 4. New York, JTS 2633 (= EMC 797; F 28 886), 16th c., 7 pp. (Part I, §§2–3). 5. New York, JTS 2639 (= ENA 1576; F 28 892), copied in 1478 for Caleb Afendopulo (see the colophon, quoted below), one of Komtino’s students, during the author’s lifetime. Afendopulo added notes in the margin. It consists of 80 folios (21×15 cm), but two of them (1, 64) are missing. Appended to the text is an oath not to reveal the contents, evidently in Afendopulo’s hand, and thus not part of Komtino’s work.26 Unfortunately this extremely valuable copy—the oldest and, as we will see, probably the archetype of the textual tradition—is badly damaged and some passages are illegible. 6. New York, JTS 2696 (= ENA 2785; F 28 869), 19th c. 7. Oxford, Bodleian, Hebr. d5 (Neubauer 2774; F 22 729), copied in 1522, fols 1a–17a (Part I only) 8. Paris, BNF, hébr. 1031/3 (F 15 723), 15th–16th c., fols. 26a–64b (part on geometry, fols. 42–64). This MS is damaged from the middle of the third section on geometry, which breaks off with a problem on the measurement of a cone. 9. St. Petersburg, Firkowitz, Evr I 343 + 344 (F 50 961), 1484–85, 32 fols. The section on geometry begins on fol. 17b. The end of the codex is damaged at the very start of Part III, chapter 1, of which little beyond the title survives. 10. St. Petersburg, Firkowitz, Evr I 320b (F 50 984), 1495, fols. 1–40 (part on geometry on fols. 19a–40b). One folio, between the current 35 and 36, is missing, and with it part of §3 on the measurement of solids.
Moritz Silberberg published the first detailed description of this treatise in 1905–1906, based on Berlin Or. Qu. 308 and St. Petersburg Firkowitz 343.27 When Pincus Schub returned to the work in 1932, he referred to six libraries with manuscripts (Berlin, Paris, Leningrad [St. Petersburg], the British Museum [Library], the Bodleian, and the Jewish Theological Seminary of New York). He
26 See Pincus Schub, “A Mathematical Text by Mordecai Comtino (Constantinople, XV Century),” Isis 17(1) (1932): 54–70, on pp. 69–70. 27 Moritz Silberberg, “Ein handschriftliches hebräisch-mathematisches Werk des Mordechai Comtino (15. Jahrhundert),” Jahrbuch der Jüdisch-Literarischen Gesellschaft 3 (1905): 277–292; II., Jahrbuch der Jüdisch-Literarischen Gesellschaft 4 (1906): 214–237.
188 Tony Lévy and Bernard Vitrac based his study on New York, JTS 2639 (Adler 1576), without noting that this is in fact the Amsterdam manuscript catalogued by Steinschneider.28
On the basis of the colophon at the end of the work, Schub was able to give :(ספר החשבון והמדות) it a title: The Book of Calculation and Measurements
Here ends the Book on the Science of Number and the Science of Measurement written by our master, the scholar, Master Mordekhai may God protect him and grant him life, the son ,(כומטיינו) Komtiano of Eliezer Komtiano, may his memory be blessed. It was copied by Shabbetai the son of Moses of Candia in the year 5238, on the 24 of the month of Adar [=Feb. 27, 1478]. ... I copied it for Rabbi Caleb, known as Afendopulo, the son of Elijah the son of Judah, his soul in paradise.29
Silberberg described the contents of Book II (on geometry), which has four main subdivisions: (1) the measurement of plane figures; (2) the divisions of plane figures; (3) the measurement of solid figures; (4) the divisions of solids. He provides a number of excerpts, translates some of them, and recasts several problem sets in modern mathematical language. Our examination of the Berlin manuscript has persuaded us that Silberberg’s description, although faithful, is not exhaustive. It refers to what he deemed to be the most important sections and provides an overview of the entire book, but a number of sections, which he evidently considered too elementary, are omitted. Silberberg noted Komtino’s
28 See Schub, “A Mathematical Text,” pp. 55–56, n. 6. Steinschneider merely noted the existence of a copy in the Lehren collection in Amsterdam, with no further information. That this copy is now the New York codex was recently demonstrated. See Benjamin Richler, Guide to Hebrew Manuscripts Collections, 2nd ed. (Jerusalem: The Israel Academy of Sciences and Humanities, 2014), p. 242. 29 New York, JTS, MS 2639, fol. 72b. Firkowitz Evr. I 320b, also from the end of the fifteenth century, concludes with the following colophon: “Executed by the young Moses, son of Abraham Ṣahlun; the work was completed on Friday, 15 Iyyar 5255 (= May 9, 1495) [in fact, this Hebrew date corresponds to Shabbat, not Friday!]; I executed it for my own use and God has opened my heart to understand all the matters of this science and to give me knowledge.” The title is not specified. The ends of Firkowitz Evr. I 343+344 and Paris are damaged. In the former, at least, the colophon of the arithmetic section survives on fol. 17b, and bears the date 1485, the Berlin codex lacks a colophon.
189 Hero of Alexandria and Mordekhai Komtino references to Euclid, Nicomachus of Gerasa, and Ptolemy, but did not identify the main sources of this book on geometry.30 Schub did so in his article (“A Mathematical Text,” pp. 56–57), written 25 years later. The main sources of the section on geometry are the Treatise on composed in Hebrew (חיבור המשיחה והתשברת) Measuring Surfaces and Volumes by Abraham Bar Ḥiyya in the first half of the twelfth century, and some Greek metrological works by Hero of Alexandria (first-second centuries CE). Schub notes that Komtino does not actually name these authors. With regard to the Greek sources, Schub delved further: Komtino employed MS Seragliensis G.I.1 (second half of the tenth century, designated S below),31 which is the only codex that contains Hero of Alexandria’s Metrica.32 The identification of S as one of Komtino’s two main sources is Schub’s most
30 Of course Silberberg, in 1905–1906, could not make use of Heiberg’s editions (1912, 1914, 1927). Strictly speaking, he could have known of the (re)discovery of the Greek codex S and have taken notice of the recent edition of the Metrica (Schöne, 1903). For the pseudo- Heronian section, he could have had recourse to the older edition by Hultsch (Heronis Alexandrini Geometricorum et Stereometricorum reliquiae [Berlin: Weidmann, 1864]). Still, because Hultsch was unaware of the existence of S, the possible textual correspondences with Komtino would probably have eluded him. As to the other main source, Abraham Bar Ḥiyya, Guttmann’s edition was not published until 1912–1913, although the Latin version by Plato of Tivoli (slightly different from the Hebrew) was available, thanks to Maximilian Curtze (Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance [Leipzig: Teubner, 1902]). In fact, a scholar with knowledge of Hebrew, like Silberberg, could have knowledge of the original from the manuscripts. 31 At the time (and until recently) designated the Constantinopolitanus Palatii Veteris 1 and dated from the 11th century. A photographic reproduction was published in Codex Constantinopolitanus Palatii Veteris No. 1, edited by E.M. Bruins, Janus Supplements volume II, 3 vols (Leiden: E.J. Brill, 1964), vol. 1. Since then, the copyist of the manuscript has been identified: it was the monk Ephrem and the date of the copy can thus be placed around 950. This codex is a monothematic manuscript that is the oldest surviving witness of the ancient Greek metrological corpus. For a detailed description of the manuscript, see: Héron d’Alexandrie, Metrica. Introduction, édition critique, traduction française et commentaires par F. Acerbi et B. Vitrac (Pisa: Fabrizio Serra editore, 2014), pp. 85–97. 32 The treatise was identified in S in 1896 by Richard Schöne and published by his son Hermann: Heronis Alexandrini opera quae supersunt omnia. Volumen III. Rationes dimetiendi et Commentatio dioptrica, recensuit H. Schöne (Leipzig: B. G. Teubner, 1903).
190 Tony Lévy and Bernard Vitrac important contribution, even though his supporting argument is rather sketchy. Because he devoted more space to the section on arithmetic (ten pages) than the section on geometry (only one page), Schub arrived at several assertions that are in fact problematic. Thus Schub seems to hold that all the texts that Komtino took from codex S are by Hero. In fact, this is not the case. We must distinguish the Metrica from the many other collections of problems (which for convenience’s sake we will designate “pseudo-Heronian”) that Heiberg classified under four headings: Geometrica, Stereometrica I, Stereometrica II, and De mensuris, and printed in volumes 4 (1912) and 5 (1915) of his edition of Hero.33 But some of the items in these collections exist in multiple versions;34 simplifying somewhat, we must distinguish at least two main textual families: that of S, and that of a large group of Greek manuscripts, of which the oldest are Parisinus gr. 1670 (twelfth century) and Parisinus suppl. gr. 387 (early fourteenth century), designated A and C in Heiberg’s apparatus. To this we can add Vaticanus gr. 215 (V), which agrees sometimes with one and sometimes with the other of these two families.35 Given the fact that there are multiple versions of the pseudo-Heronian material used by Komtino, we need to delve more deeply in order to identify his source: S or the other family? Schub did not think this worth the effort, but we will demonstrate that Komtino took his pseudo-Heronian material from S. We have another reservation about Schub’s argument: his firm conviction that Komtino used codex S rests on an assumption we do not believe to be adequately grounded. According to Schub, Komtino made rather flexible use of his Hebrew source,
33 In the Greek manuscripts, collections of problems corresponding to these titles are sometimes attributed to Hero (such as the De mensuris) and sometimes not. Often they are repetitive and incoherent, which rules out their having been the work of a single author, whoever he may be. They are the product of repeated contaminations. Those that Heiberg collected and entitled Geometrica and Stereometrica I are his own artifacts. See Héron d’Alexandrie, Metrica, pp. 449–474. 34 Unlike the Metrica, transmitted only in S, there are around a hundred manuscripts of collections (or portions of) of the pseudo-Heronian corpus (although often of a late date). 35 Codex A contains tables, a formulary, and a series of problems in plane geometry; codex C preserves, in addition, the set of definitions attributed to Hero and a series of problems in stereometry. As for V, it is a collection of extremely chaotic compilations, traditionally referred to as the Liber geeponicus. See Héron d’Alexandrie, Metrica, pp. 434–471, 491– 492, and 581–582.
191 Hero of Alexandria and Mordekhai Komtino
Bar Ḥiyya, but translated his Greek literally, reproducing all its details, including errors36 and diagrams.37 We will show that this is not the case: Komtino made a creative use of his Greek source, which is, however, codex S, as Schub maintained. *** Aside from the two pioneering studies by Silberberg and Schub, Komtino’s treatise has been ignored. In 2014, a new critical edition of Hero’s Metrica, with French translation and commentaries, was published by Fabio Acerbi and Bernard Vitrac (supra, n. 31); it included a long essay by the latter on the legacy of the Metrica (Héron d’Alexandrie, Metrica, pp. 429–588). In fact, the Greek treatise was referenced only twice in late Antiquity, by Eutocius and by the author of the anonymous Prolegomena to the Almagest; that is, by two figures who belonged to the same learned circle (that of Ammonius) in fifth- and sixth-century Alexandria.38 Other texts by Hero, notably some of his works on mechanics (the Mechanica and Pneumatica), were translated into Arabic and Latin and made his reputation. His commentary on Euclid’s Elements—the earliest known—was translated into Arabic and quoted by a number of authors. But no medieval Arab or Latin author mentions the Metrica and we do not know of any translation into these languages. So far as we know, codex S never left Constantinople.39 So it is far from insignificant that in the fifteenth century, in Constantinople, a learned Jew consulted and drew on this manuscript. This gave Heron’s Metrica and the pseudo-Heronian material a (Greco-)Hebrew posterity—the only known legacy of the Metrica. The unlikely encounter between Komtino and the
36 He mentions two of them, borrowed from exercises in the stereometric section. See Schub, “A Mathematical Text,” p. 69. 37 Ibid., p. 57 n. 11 38 See Héron d’Alexandrie, Metrica, pp. 23 and 127–128, respectively. 39 It is likely that it was never copied, even if it was read by (at least) three persons: Seven marginal annotations in codex S itself, in an unidentified twelfth-century hand, referring to collections of problems of the pseudo-Heronian corpus, mention Hero’s demonstrations and clearly allude to propositions in the Metrica (I.8, 21, 32, 33; II.3); one of them cites lemma I.18/19 in full. Also in S, there are several sparse and brief annotations in a hand that has been identified as that of the erudite Maximus Planudes (end of the thirteenth century). And as we will show, Komtino also used it.
192 Tony Lévy and Bernard Vitrac ancient Greek metrological corpus, both Heronian and pseudo-Heronian, quite unique, deserves to be brought to the attention of readers of the new edition of the Metrica. A mere reference to Schub’s article does not seem to be adequate, given its brevity and the weakness of some of his assertions. The present authors wanted to study the modes and methods of this encounter and undertake a detailed comparison of the Greek and Hebrew texts. A summary of our initial results was incorporated in the aforementioned edition of the Metrica.40 The present article goes into the subject in much greater detail. We should make it clear that we deal here with the geometrical part of Komtino’ s work (henceforth Komtino’s geometry) only. Our object is neither a full analysis of all the sources of this Hebrew text nor the preliminaries for a critical edition. Also, of the eight manuscripts mentioned above that contain Komtino’s geometry in whole or in part, we made systematic use of only four: New York, JTS 2639 (Adler 1576), St. Petersburg Firkowitz Evr I 320b, Berlin Or. Qu. 308, and Paris, BNF, héb. 1031/3. In several places we consulted three others: London, BL Add. 2717 Alm. 213, St. Petersburg, Firkowitz Evr I 343+344, and New York, JTS 2696 (Adler 2785).41 The two most recent copies of the work (from the nineteenth and twentieth centuries), which we consulted occasionally, are interesting for the history of the text but do not seem to shed any light on its production by Komtino. With regard to the Greek sources, we provide the information required for an independent reading of this article, even though some of it can be found in the edition of the Metrica. It goes without saying that interested readers will find there additional details about the ancient Greek metrological tradition. We begin with a description of the overall structure and content of the geometrical part in Komtino’s work. Then we analyze each of his borrowings from the ancient Greek metrological corpus. In addition to the content, we are interested in how Komtino approached these different versions. We cannot totally accept Schub’s thesis that Komtino adhered to his Greek source verbatim, translating blindly and reproducing all its details, including the mistakes. In fact, Komtino’s method depends on the themes discussed; it also varies according to whether he is reproducing definitions, proofs or demonstrations, algorithms, or computations.
40 Héron d’Alexandrie, Metrica, pp. 549–556, esp. p. 550. 41 We use the following sigla/abbreviations to designate them: NY, Pet., B, P, L, Pet. 343/4, NY 2696, respectively.
193 Hero of Alexandria and Mordekhai Komtino
We start with an explanation of our method and objective—an objective that one may deem to be modest or indeed inadequate to justify an account of such dimensions. We do not think so. Our study belongs more to the domain of the history of texts than to the general study of social contexts or the intellectual history of the Jewish community of Constantinople in the fifteenth century. We describe the indirect Hebrew tradition of the Metrica and the pseudo-Heronian corpus and undertake a minute comparison of the texts, particularly to see how a Byzantine author drew on certain traditions of classical Antiquity. What we propose here is a case study of a widespread phenomenon in the Byzantine world, specifically a case in which we have access to the specific ancient model (codex S) and to important manuscripts of Komtino’s geometry copied soon after it was written. This makes it possible to document precisely the author’s borrowings as well as his biases, some of them innovative. In this way, readers will learn about a category of works that deserve to be better known but which generally have little chance of escaping oblivion. In most cases, these Byzantine writings are seen as no more than byproducts of the decay of the classical tradition, devoid of originality and not worth spending time on. We hope that our case study, limited but detailed, will make it possible for others to describe, in a broader historical perspective, these phenomena of inheritance and borrowing in the Byzantine world in general and in Komtino’s milieu in particular. The article has three appendices. The first, on the structure of Komtino’s geometry, provides a detailed table of contents of the work; the other two endeavor to clarify the relations among the various Hebrew manuscripts of the work. This allowed us to gauge the reliability of our textual witnesses and their proximity to the original. For example, MS New York is the closest and MS Paris much less so. In the absence of a full stemma, these comparisons may facilitate the work of future editors of this treatise.
The geometrical part of Mordekhai Komtino’s work Part II of Komtino’s work42 begins with the following subtitle (Figure 1):
Book Two of this treatise, which deals with the science of measurement. The first chapter of the first section of this second book.
42 NY, fols. 30b–72b; Pet., fols. 19a–40a; B, fols. 36a–95a (= pp. 71–189); P, fols. 42a–64b; L, 44a–80b.
194 Tony Lévy and Bernard Vitrac
Mordekhai said: Before dealing with the science of measurement it is appropriate to explain the names employed by the masters of this science because, in each science, specific names are used by its masters.
Figure 1 (MS JTSL 2639, fol. 30b) Courtesy of The Library of the Jewish Theological Seminary
As has been stated, Komtino’s geometry is divided into four sections: (1) the measurement of plane figures, (2) the divisions of plane figures, (3) the measurement of solid figures; (4) the divisions of solids. Each section is in turn divided into chapters. For example, the first section has six chapters:
1.1 The terminology of plane geometry 1.2 The measurement of squares, rectangles, and their diagonals 1.3 The measurement of triangles, according to their several species according to their (תמונות נוטות) The measurement of slanted figures 1.4 several species43 1.5 The measurements of circular figures 1.6 The measurement of regular polygons
The general arrangement of Komtino’s geometry in four sections, the choice of the sections and subsections—in brief, the book’s structure44—is taken over from Abraham Bar Ḥiyya’s Treatise on Measuring Surfaces and Volumes. A fair
43 Hero uses the same Greek work, , for all figures of this description, both quadrilaterals with two parallel sides (see Metrica I.11–13) and those with no parallel sides (see Metrica I.14-16). Komtino, too, uses a single term for both. For the first category, we translate the Greek and Hebrew as “trapezium”, and render the second species “quadrilaterals.” 44 See Appendix A.
195 Hero of Alexandria and Mordekhai Komtino amount of that work was borrowed, rewritten, paraphrased, or abridged by Komtino. But within this general framework, there is also material of Heronian or pseudo-Heronian origin on trapezia, circles, regular polygons (from the pentagon through the dodecagon), and pyramids. The Heronian borrowings are confined to sections 1 and 3, which deal respectively with plane and solid figures. The table lists the Heronian and pseudo-Heronian borrowings:
Heronian and Komtino (NY) pseudo-Heronian Editions Content corpus (S)45 fols. 31b–32b fol. 4r–v 46 Terminology; list of plane figures §1.1 Geom. 3.1–23 fols. 40b–44a Measurement of trapezia and fols. 73r–76r Metrica I.11–16 §1.4 quadrilaterals
fols. 44a–48a fols. 9r–10r, 11r, Geom. 17.4bis–3bis Measurement of circles and their §1.5 12r (?) Geom. 19.6, 8 segments Procedures for finding the area 47 fols. 48a–54a fols. 19r–20v Diophanes 10–17 of regular polygons Pn (5 ≤ N ≤ 12)
45 In this column we indicate the location in S of items taken over by Komtino. Some of them appear in other manuscripts of the Heronian corpus. Did Komtino take them exclusively from S? We will show that this is indeed the case. 46 For convenience’s sake, we sometimes use the designations employed by Heiberg in the Teubner edition (Geom. = Geometrica, Stereom. = Stereometrica). See above, n. 33. 47 On folio 17v of codex S there is a reference to a certain Diophanes, corrected by a later hand to “DiophanTus,” in a heading that marks off a new and rather chaotic section (17v16–26v25); see Héron d’Alexandrie, Metrica, pp. 481–488. This collection of problems also exists in another manuscript, Parisinus gr. 2448, fols. 70v23–76v19, with some variants, and is attributed there to Diophantus. Consequently, it was included by Paul Tannery in Diophanti Alexandrini opera vol. II (Leipzig: B.G.Teubner, 1895), pp. 15.20–31.22. Heiberg edited it again, with special attention to S, in Mathematici Graeci Minores XIII,3 (Copenhagen: Bianco Lunos Bogtrykkeri, 1927), pp. 26–64. There is a French translation in: Paul Ver Eecke, Les opuscules mathématiques de Didyme, Diophane et Anthémius suivis du fragment mathématique de Bobbio (Paris and Bruges: Desclée de Brouwer, 1940), pp. 19–44. Hence it is referred to as Diophanes or pseudo- Diophantus.
196 Tony Lévy and Bernard Vitrac
Measurement of these polygons §1.6 fols. 77v–81r Metrica I.17, 18–25 (and the equilateral triangle) with proofs fols. 63b–67b Stereom. II.55, Measurement of pyramids fols. 55v–61r §3.2 II.57–58, 60–68 and truncated solids Measurement of spheres and fols. 67b–71a fols. 12v–13v Stereom. I.55–60 portions Stereom. I.19, 12, Measurement of cylinders, §3.3 fols. 14r–15r 18, 15 cones, conical frustra
None of the material about the divisions of figures (sections 2 and 4) has a Heronian origin, so Komtino could not have taken it from Book III of the Metrica (S, fols. 99v–110v), which deals with such questions (along with other criteria). In fact, Komtino is interested in the divisions of figures into two, three, four, or more equal parts, whereas all of the 22 problems presented by Hero, with two exceptions,48 deal with the divisions of figures according to a specified ratio; what is more, the solutions given there, unlike those to the problems in the Hebrew treatise, are analytical. This means that Komtino took his discussion of the divisions of figures from other authors, especially Abraham Bar Ḥiyya, and perhaps other sources (who these might be remains to be verified, but is not part of our objective here).
The First Encounter: The Terminology of Plane Geometry Quite naturally, Komtino begins his geometry with a discussion of terminology. Like many before him, he does so through a free paraphrase of the definitions in Book 1 of Euclid’s Elements.49 He also has to introduce the notion of “measurement”—the subject of this part of the work—which he does in a passage whose inspiration cannot really be Euclidean:50
48 Metrica III.18, 19: division of a circle (or triangle) by two chords (or three straight lines running from an interior point) into three equal parts (that of the circle is approximate); these problems were not reproduced by Komtino. 49 The author and work are cited explicitly. Komtino also includes the definition of the segments of a circle, originally introduced in Book III (def. III.6) but frequently moved so as to be closer to those of the circle and semi-circle (deff. I.15–18), including in most Greek manuscripts of the Elements. 50 The notion of “measurement” ( ) certainly exists in the Elements; one could even say that it plays a key role there. But it is presented according to a theoretical approach
197 Hero of Alexandria and Mordekhai Komtino
The area of a figure [is obtained by taking] the right-angled quadrilateral parts whose length is the same as their width or [the parts] that are equal to them [in area] and that are needed to fill in and overlap the figure. And the [corresponding] operation is the product of one side by another, or of a straight line by a curve, or of a number by a number.
Komtino continues with a new and rather curious mention of Euclid, introducing a text on geodesy that contains a second part on terminology, and beginning as follows (NY, fol. 31b):
וכבר באר אקלידס בספר אחד שחבר במדידת הארץ Euclid explained in a book that he wrote about שחכמת המדות השטוחה היא מיוסדת מארבעה דברים, the measurement of land that the science of the מהפאות ומהנקדות ומהקוים ומהזויות, ומקבלת גם measurement of plane surfaces is based on four סוגים ומינים. things: directions, points, lines, and angles. And it also has species and types. והפאות הם מזרח ומערב צפון ודרום. .The directions are east, west, north, and south והנקודות הם אשר תקח אותם ה]..[ח]..[ה או סימן The points are those that you select [...]52 as the לדבר. .sign of the thing והקוים הם עשרה, קו ישר ותושבת וראש ועמוד וקו ,There are ten lines: straight line, base, top נכחי וקטר ושוקיים ומקיף ואלכסון ומיתר. ,perpendicular, parallel line, diagonal, [two] legs perimeter, diameter, and hypotenuse.
whose ultimate goal is to distinguish the lines and areas that can be expressed in numbers and those that cannot be (“irrational”) and to classify the latter (in Book X), rather than to present effective procedures for measuring figures on the basis of data expressed from the selection of metrological units. Nevertheless, a number of results in the Euclidean treatise, labeled in terms of equality or proportion, can be used to validate such procedures. The same remark holds for a number of Archimedean texts; it is clear when one reads the proofs proposed by Hero in the Metrica: Euclid and Archimedes are his main “authorities.” 51 Here we follow the manuscript NY. The square brackets represent two groups, of five and two letters, respectively, that do not make sense; the fourth letter of the first group is and suppose the ת damaged (a tear in the paper); if we complete the second by reading a beginning.” If we change the first“ התחלה we could read the word as ,ל fourth might be a ”.or“ או letter in the other group, we might obtain These hypotheses take account of what seems to have been the reading of the common archetype of Pet. 343/4, Pet., B, P, L, where we find: “those that you take at the beginning or as the sign of the thing.” The current state of NY makes it difficult to see a logical disjonction. To make sense, Silberberg translated this as “those that you take as origin and extremity” (Anfangs- und Endpunkte); see Silberberg, “Ein handschriftliches hebräisch- mathematisches Werk II,” p. 216.
198 Tony Lévy and Bernard Vitrac
This is simply a Hebrew version of the lexicon found in S, fol. 4a–b, which is Geometrica 3.1–23 in Heiberg’s edition:52
1. Plane geometry is constructed of directions and points of observation and lines and angles; and it includes types and species and objects of study. 2. And there are four directions: east, west, north, south. 3. While a point of observation is each point chosen. 4. And there are ten lines: straight, parallel, base, top, legs [i.e. sides], diagonal, perpendicular [also said to be at right angles], hypotenuse, perimeter, diameter.
In this beginning, the words and indicate the relationship to geodesy, which is not encountered again, neither in the rest of the lexicon (which is purely geometrical from subsection 4 on) nor in the collections of problems. The enumeration in §2 makes the sense of (polysemic in Greek; = inclination [e.g., of a pyramid; see below], latitude, region, geographic zone, etc.) clear. On the other hand, even disregarding manuscript variants, it seems that our Hebrew text was not really sure about how to render the enigmatic (“point of observation”) in the third item of the Greek text. We must also take into account the multiple versions of this introductory text, which, as noted is found in various Greek works:
• In that referred to here, found in S, fols. 3r–17v • In Hero’s Introductions to Geometrical [Questions] ( ), a very large collection of tables and problems found in a different branch of the pseudo-Heronian tradition (codices A and C) • In the Geodesy ( ) attributed to Hero, found in other manuscripts of a later date.
52 Heronis Alexandrini opera quae supersunt omnia. Volumen IV. Heronis Definitiones cum variis collectionibus. Heronis quae feruntur Geometrica, ed. J. L. Heiberg (Leipzig: B. G. Teubner, 1912), pp. 176.15–180.23. Komtino did not reproduce 3.24–25, which deal with the enumeration of solid figures and propose “principles of measurement.”
199 Hero of Alexandria and Mordekhai Komtino
Sometimes it is reproduced in part in various works by Byzantine authors. To simplify matters, one could say there are fundamentally two versions of the text:53 that of S on the one hand, and that of all the other manuscripts on the other. In the portion employed by Komtino, the main feature in the Greek tradition that distinguishes the two versions comes in the list of eighteen plane figures (square, rectangular parallelogram, six types of triangle, rhombus, rhomboid, four types of trapezia, and four circular figures), namely, the order of presentation of the triangles: right, isosceles, equilateral, acute-angle, obtuse- angle, and scalene in S; equilateral, isosceles, scalene, right, acute-angle, obtuse- angle in the other manuscripts of the pseudo-Heronian corpus. To put it another way, in S the order is dictated by considerations of calculation; whereas in the other codices it is set by a Euclidean logic (classification of triangles by the sides, then by the angles, as in Euclid’s Definitions I.20–21, which refer, however, to trilaterals rather than triangles). Returning to Komtino, we observe that he follows the order in S, at least according to New York JTS 2639, the oldest and probably most faithful to his original text. In St. Petersburg 343/4, St. Petersburg 320b, and the Berlin,54 Paris, and London manuscripts, the order is equilateral, isosceles, scalene, right, obtuse-angle, and acute-angle, which does not accord either with S and NY or with other manuscripts of the pseudo-Heronian corpus (interchanging the last two items in the list), but is that of Definitions I.20–21 in Euclid’s Elements. Clearly there was some modification of Komtino’s text in the course of transmission, probably between 1478 (the date of New York JTS 2639) and 1484/85 (the date of St. Petersburg 343/4). It is probable that the person responsible for this change believed he was correcting an erroneous text. Given that Komtino began his exposition of geometry with a terminological discussion inspired by the definitions in Book I of the Elements (albeit freely paraphrased), in which he listed the types of triangles in Euclid’s order, but then switched to a different source (which we know was S, fol. 4 = Geometrica 3.1–23), it was
53 In fact, we can distinguish three versions of this frequently copied text, but the other differences play no role here. Interested readers are referred to Héron d’Alexandrie, Metrica, pp. 455–461. 54 Consequently, this is what is given by Silberberg, “Ein handschriftliches hebräisch- mathematisches Werk II,” pp. 216–217.
200 Tony Lévy and Bernard Vitrac perfectly reasonable for the copyist to think that order of the second list (which actually derived from a different model and followed a different logic) was corrupt and should be set aright. There are other variants that distinguish the two Greek textual families, in the definitions of the straight line (Geometrica 3.5),55 base (Ibid. 3.7), “linear” measurement (Ibid. 3.19),56 the list of rectilinear angles (Ibid. 3.15),57 and the list of quadrangular figures (Ibid. 3.23).58 In all these cases, Komtino’s text agrees with S or, where it diverges slightly, presupposes such a text. Since there is no doubt that Komtino was following a pseudo-Heronian text in codex S, why did he invoke Euclid’s authority? Schub proposed an explanation for this misstep, based on a peculiarity of the beginning of that codex. In the several known versions of the text we are discussing here, it is often preceded by the title (Hero’s introduction to geometrical [questions]). But the title is not found in S. Worse still, the text was originally headed by a (now illegible) heading that was scratched out (see Figure 2), after which a later hand wrote (Euclid’s geometry).
55 (S: “is ‘straight’: the line stretched in a straight line whose ends are points”) versus (VAC: “is ‘straight’: the line that is a straight line”). 56 (S: “‘Linear’ is then everything that is measured rectilinearly, that has only length, which is also called ‘number’”) versus
(VAC: “‘Linear’ is then everything that is measured rectilinearly, that has only length, which is then called both ‘principle’ and ‘number.’” 57 S: (“And the angles are [of] three [types]: right, acute, obtuse”); VAC: (V: ) (“right, obtuse, [and] acute”). 58 S: (“2 quadrangular objects of study: quadrangle, rectangular parallelogram; VAC:
(“2 quadrangular objects of study: rectangular equilateral quadrangle and rectangular equilateral parallelogram”). Komtino evidently follows S, whence the absence of the square, which, in principle, should head off the list. In NY, a later hand notes that the quadrilateral with parallel sides and right angles is an oblong.
201 Hero of Alexandria and Mordekhai Komtino
Figure 2: MS Seragliensis G.I.1, fol. 3r apud Bruins (Brill, 1964, supra note 31), p.1
This is followed by a double preface and a table and then (on fol. 4r) the text that we have just discussed. Komtino could have credibly believed he was dealing with a treatise by Euclid when he began translating Geometrica 3.1–23 from S. This explanation is persuasive, but of only local validity. One cannot deduce from it, as Schub seems to have done, that, because he does not mention him, Komtino failed to identify Hero among his sources.59 As we shall see, Komtino drew on several of the treatises found in S (“scanning” the manuscript as it were), and Hero is mentioned there several times, in the titles and subtitles of the Metrica (fols. 67r, 87r, 99r) and in another metrological introduction on fol. 27r (Euclid, too, incidentally, is here, on fol. 3r, and for a collection of metrological formulas on fol. 61r). So if Komtino does not refer to Hero, it may be because he believed his readers were familiar with Euclid, as well as with Ptolemy and Nicomachus of Gerasa, but not with Hero. Nor does he mention Diophan(t)us (S, fol. 17; see above, n. 47), satisfied with a reference to the “book of the ancient wise men,”60 with no further detail, when he applies his procedure for regular polygons. Furthermore, Komtino certainly understood, even if in hindsight, that he was working from a geometrical treatise by Hero. This is confirmed by his remark on Proposition II.14 of Apollonius’ Conics in his commentary on Maimonides’ Guide: “This book [the Conics] has not come into our hands and we have not seen it.
59 Schub, “A Mathematical Text,” p. 57. Nor does he refer to Abraham Bar Ḥiyya; according to Schub (p. 58), this is because he confused him with his “homonym” Abraham Ibn Ezra, already quoted in the first (arithmetic) section of the work. 60 “Know that the figure with straight lines can be broken down into triangles and the is ,( בספרי החכמים הקדמונים) triangle itself, as was shown in the books of the ancient sages the basis and root of all rectilinear figures...” Thus the text in NY, B, L, and New York .(is missing in Pet. 343/4, Pet. (fol. 29a15) and P (fol. 52b22 החכמים ;2696
202 Tony Lévy and Bernard Vitrac
What has come to our hands, however, is one of the Greek books in this domain and contains many themes that escape (אירון) geometry]: it was written by Hero] the imagination and that [only] proofs bring to light.” 61 Even if this passage does not refer explicitly to the Metrica, we know that only two works on geometry were attributed to Hero, the Metrica and his commentary on Euclid’s Elements, which does not seem to be pertinent in this context. Another quibble with Schub’s theses: as we have mentioned, he holds that whenever Komtino’s source was Greek he merely reproduced S in literal translation.62 This is not the case here. In the list of the ten types of lines (see above, No. 3), all the versions of the Greek manuscripts follow the same order, but, as shown here, Komtino’s order does not correspond to that of the Greek:
Order in the Greek Komtino’s Order
Straight line, parallel, base, top, legs, diagonal, Straight line, base, top, perpendicular, parallel perpendicular (also called at right angles), hy- line, diagonal, [two] legs, perimeter, diameter, potenuse, perimeter, diameter hypotenuse
We should also note that the explanations provided for each of these terms, both in the Greek text (Geometrica 3.5–3.14) and in Komtino’s translation, stick to the order of the associated list. This is not the only one that was changed: an entire textual unit was altered, even if the principle behind the order eludes us. In addition to the change in the order, there are also local corrections to the text in some of the explanations: Komtino simplified the description of the base (No. 7, slightly corrupt in S) and filled in the scanty description of the diameter (No. 14). Emerging from this first encounter, we thus have strong indications that Komtino used codex S for the first section of the first book of his geometry, as well as reasons to think that this use was not merely a compilation. We have also observed
.הרון מאלכסנדרי .See the index of sources in Schwartz, “Understanding in Context,” p. 170, s.v 61 We are grateful to Ruth Glasner (Jerusalem) who brought this reference to our attention, and to Gad Freudenthal, who sent us the present passage of Komtino’s manuscript. The Hebrew text of Komtino’s commentary has recently been edited and published: Esti Eisenmann and Dov Schwartz, The Commentary of Rabbi Mordekhai ben Eli’ezer Komtiyano on Maimonides’ Guide of the Perplexed (Ramat Gan: Bar Ilan University Press, 2016) (Hebrew). For the above mentioned quotation, see there, pp. 234–235. 62 Schub, “A Mathematical Text,” p. 68.
203 Hero of Alexandria and Mordekhai Komtino that the text of Komtino itself has been modified in the course of transmission, requiring some caution on our part and inviting us to continue our investigations.
(הנוטים) The Second Encounter: The Measurement of Slanted Figures In the subsections that he devotes to the measurement of squares, rectangles, and triangles, Komtino derives free inspiration from Bar Ḥiyya’s Treatise on Measuring Surfaces and Volumes, with occasional incongruities,63 but evidently with no reliance on any Greek metrological text. Note that he does not follow Bar Ḥiyya when the latter proposes (§7364) a general method for calculating the area of triangles without using their height (but for the trivial example of the right triangle [6, 8, 10]) by the so-called method of remainders, which is valid for every type of triangle and corresponds to Hero’s celebrated formula: A =√ [ p ∙ (p – a) ∙ (p – b) ∙ (p – c)], where (a, b, c) designate the lengths of the sides and p is the semiperimeter. Bar Ḥiyya recognized that he could not provide a proof at this juncture of his text; and because Komtino seems to want to ground his statements as much as possible, he skipped over the famous procedure. Ironically, he could have found the proof in his other source, S, inasmuch as that is the topic of Metrica I.8. The procedure changes in the subsection on the measurement of the slanted figures. Komtino, following Bar Ḥiyya, could have distinguished three types of quadrangles that are neither equilateral nor rectangular: non-rhombic parallelograms (§74), trapezia (§§76–93), and any quadrilateral (§94). But this is not what he does: first he considers only trapezia, distinguishing the different types but taking over only the first example of an isosceles trapezium (18, 8, 13, 13) from his predecessor (§§76–78). But he soon drops this to insert a set of problems that correspond to most of Metrica I.11–16. The contents are as follows: Metr. I.11: Height (designated the “perpendicular”) and area of an isosceles trapezium ( = = 13, = 6, B = 16). Metr. I.12: Height (designated the “perpendicular”) and area of an acute- angle trapezium ( = 13, = 20, = 6, B = 27). Metr. I.13: Height (designated the “perpendicular”) and area of an obtuse-
63 See below, Appendix A, especially on subsections 1.3.2–1.3.4 (nn. 143-145). 64 This division is that of the Hebrew text translated by Millàs: Michael Guttmann, Chibbur ha-Meschicha weha-Tischboreth (Berlin: Mekise Nirdamim, 1912), in Abraam Bar Hiia, Llibre de geometria. Hibbur hameixihà uehatixbòret, trans. J. Millàs i Vallicrosa (Barcelona: Editorial Alpha, 1931).
204 Tony Lévy and Bernard Vitrac
angle trapezium ( = 13, = 20, = 6, B = 17). Metr. I.14a: Remarks about rhombuses. Metr. I.14b: Quadrilateral with right angle but no parallel sides, = 13, B = 10, = 20, = 17. Demonstrate that its area is given. Metr. I.15 (certainly spurious): Prove, for the same figure, the last sentence of Metr. I.14b: “The [straight line] drawn from perpendicular to is given.” Metr. I.16: Another quadrilateral of the same type, with right angle but no parallel sides, = 13, B = 10, = 8, = 25. Find its area.
The first two books of Hero’s Metrica present algorithms for calculating the area of plane figures (Book I) and the volume of solids (Book II), accompanied by geometric proofs. We shall consider the example of Metrica I.12 (Héron d’Alexandrie, Metrica, pp. 176–177; NY, fol. 41b–42a):
Hero Komtino
XII Suppose an acute angle trapezium If you wish to measure a trapezium with acute with angle acute, and let the line be 13 angles like trapezium ABGD, and the angle at (אמות) units ( ), line 20 units, and 6 side B is acute, and side AB is 13 cubits units, and 27 units. Find the perpendicular and side GD 20 cubits and side AD 6 cubits and the area. and side BG 27 cubits, and if you want to find the height and the area, proceed as follows.
K A ] [ ] [ ]M[
] [ H ]K[
B A Z E N
(see below) You subtract the 6 cubits from the 27 cubits, leaving 21 cubits, and because the sides of the namely ,(ידועות) acute-angle triangle are known 13 and 21 and 20, one also knows the length of the height, which is 12 cubits, as we have .(כמו שידענו זה מקודם) learned before
205 Hero of Alexandria and Mordekhai Komtino
Hero Komtino
And to the 6 cubits we add 27 cubits, and half of the result is 16½ cubits;
We multiply this by the height: this gives 198 and this is the area of the trapezium.
And the proof of this measurement is:
Let a straight line be drawn parallel to I draw line AE parallel to line GD and take and a perpendicular straight line . AZ as the height.
On the one hand, will then be 20 units, and Given that AE is 20 cubits, GE is 6 cubits, thus 6 units: the remainder is 21 units; BE which remains is 21 cubits. such that—because triangle is acute This is why, because line AZ, the height of the angled—the perpendicular is 12 units. acute-angle triangle ABE is 12 cubits,
Now because sides and are divided in And the sides ABGD are bisected by points H two [equal parts] at [points] , , and [lines] and T and because KHL is thus perpendicular , , being perpendicular, to MTN, similar to the preceding, we will demonstrate Just as we have demonstrated in the previous ( ) that, on the proposition, so too it will be demonstrated (כמו שהראנו בצורה הזאת הקודמת, לזה יראה) one hand, trapezium is equal to parallelogram , that trapezium ABGD is equal to the area of KLMN, [with] parallel sides. and on the other hand, that , one with Now line (BG AD), taken together, is twice the other, is twice . And is 16½ units; KM, and KM is 16½ cubits. And KL is 12 but in addition, is 12 units—because also cubits, because it is the same as line AZ. —the area of the trapezium will thus be 198 If so, the area of the trapezium is 198 cubits. units.
And in consequence of the analysis, this will be synthetised as follows ( ; ).
Subtract 6 from 27: the remainder is 21. (See above)
And the sides of an acute angle triangle being given ( ): 13 and 21 and 20, find the perpendicular ;
It is 12 units, as we have already learned ( ).
Now add 27 and 6: the result is 33, half of which is 16½; [multiply] this by 12: the result is 198. And this will be the area.
206 Tony Lévy and Bernard Vitrac
In much of his treatise, Hero follows the same two approaches for each problem:65 validation of a procedure by means of a geometrical analysis that employs the Euclidean terminology of givens, and an implementation of that procedure in a “numerical synthesis.”66 This double approach astonished Komtino, who does not seem to have approved of it. Consequently, in his equivalents of propositions I.12, 13, 14, and ,(ידועה :of the Metrica, even though he sometimes retains the givens (Hebrew 16 Komtino reverses the order of the numerical procedure (‘synthesis’) and analytic proof, according to the following scheme:
Metrica Komtino
Geometrical analysis in terms of givens ——
—— Numerical example
Introductory formula of the synthesis Introductory formula of the proof
Numerical synthesis Geometrical proof
Komtino does not repeat the Heronian formula that introduces the numerical procedure after the analysis,67 of course, but he does insert it for the proofs: “And the proof of this measurement is.” He also deletes the short section on rhombuses and rhomboids (Metrica I.14a), which is out of place in a chapter on trapezia; this forces him to slightly
65 Sometimes the procedure is simpler and Hero deduces the computation directly from a result of theoretical geometry, cited most often with a reference along the lines of “and because it has been demonstrated by Archimedes in the Method that ...” On the variety of Heronian methods in the Metrica, see Héron d’Alexandrie, Metrica. 66 The method of analysis/synthesis is traditional in Greek geometry of the Hellenistic period (Euclid, Archimedes, Apollonius), and Hero inherited it. In the Metrica, however, he employs it in a wholly original manner and this is one of his most important innovations. It was subsequently taken up by other authors, notably Ptolemy and Pappus (see Héron d’Alexandrie, Metrica, pp. 57–59 and 363–372). 67 In I.12, 14: (“And, in consequence of the analysis, it will be synthetised this way”); in I.13, 16: (“And it will be synthetised this way”).
207 Hero of Alexandria and Mordekhai Komtino recast the start of the problem in Metrica 1.14b. He maintains the Heronian order in Proposition 1.15 (certainly spurious), which is not a problem about measuring the area of a quadrilateral but a proof that a certain height, used in Metrica 1.14b, is indeed given. Thus this is the only case where Komtino employs the equivalent of the Heronian formula that introduces a numerical procedure, in ”.(אופן המעשה כן) the form, “and this is how it is done This second encounter confirms our earlier impression: unless we suppose that Komtino had access to another manuscript of the Metrica that has since vanished without a trace, we have additional evidence that Komtino employed codex S,68 but allowed himself a certain latitude rather than simply translating word for word.
The Third Encounter: Measurement of a Circle This section begins by reviewing the definition of the circle, indicating the various types of circular figures, and laying out in general lines the structure of the section (NY, fol. 44a):
Know that the circle is a figure enclosed by a single line, with a point inside it such that all lines that run from it to the perimeter are equal. Thus this section deals with the measurement of the complete circle that we have defined and the measurement of circular segments. There are three types of circular segments. The first is the semicircle; the second is more than half the circle; the third is less than half the circle. We begin with the measurement of the complete circle, after which we will speak of the measurement of its segments.
As he specifies in his introduction, Komtino is interested mainly in measuring the area of a circle; consequently he does not highlight the algorithms69 d→circ, circ→d presented by Bar Ḥiyya after §95 and §99 and by pseudo-Hero in Geometrica 17.8a + 17.7a (at the start of a problem which Komtino had gone
68 This second encounter offers another argument, based on an analysis of the diagrams found in S and in manuscripts NY and Pet. of Komtino, that is rather complex. So as not to interrupt our description of the encounters, we refer the interested reader to Appendix B below. 69 d designates the diameter, circ the circumference, and A the area of the circle.
208 Tony Lévy and Bernard Vitrac into anyway). In a manner that is close to Bar Ḥiyya’s §95, he combines the information synthetically:
of a circle [is obtained] by (תשבורת) We say in this regard: the area multiplying half its diameter by half the line that encloses it. The ratio of the line that encloses the circle to its diameter is the ratio of 22 (ערך) 1 70 to 7. Thus the line that encloses it is equal to 3 ∕7 of the diameter. This is according to the teaching of the geometers who make precise statements about this science.
We will describe the section on the measurement of the area of the circle: Komtino presents five algorithms, along with two quotations from Ptolemy and numerical examples. As the table below shows, he took his inspiration from two sources (Bar Ḥiyya, §§95–101 and problem 17 of (pseudo-)Euclid’s Geometry71), but elaborated them quite freely.
• Komtino does not follow Bar Ḥiyya faithfully; the latter made do with only the first of the algorithms presented here (§95) and a variant of the 2 2 1 2 2 fourth (§96), d → (d) → d – ( ∕7) ∙ d – (1/14) ∙ d = A, that Komtino does not preserve. He also left out the examples of Bar Ḥiyya’s §§98–101. • He was selective with regard to problem 17 of “Euclid’s” Geometry72 and did not bring either the first mention of the algorithm d→circ or the numerical example that accompanies the first algorithm for computing the area, their proofs, or the inverse algorithms (A→ circ, circ→ d, A→ d).
70 This phrase about the ratio of circ to d (in modern terms, π) is corrupt in the four manuscripts that we consulted. (A lacuna at the start of the section on the measurement of the circle makes it impossible to know what was in Pet. 343/4). This suggests that the mistake goes back to the archetype of the surviving tradition; see below, Appendix C. 71 S, fols. 9r5–10r19; it is also found in part in another manuscript (Vat. gr. 215), which, drawing on S, transmits some excerpts from that compilation, but not in the other branch of the manuscript tradition of the pseudo-Heronian corpus (MSS AC). Heiberg edited it as 17.8a+17.7a+17.4bis+17.1+17.3+17.3bis of the Geometrica; Komtino took over subsection 17.8a+17.1+17.3+17.3bis. 72 This problem does not contain either of the two algorithms proposed by Bar Ḥiyya, although they can be found elsewhere in the pseudo-Heronian corpus.
209 Hero of Alexandria and Mordekhai Komtino
Komtino Type Algorithm Source of the algorithm
Algor. (i) (d, circ) → A (d, circ) → d/2, circ /2 → (d/2) ∙ (circ /2) = A Bar Ḥiyya 1 (§ 95)
First discussion of the ratio circ : d Geometers vs. Ptolemy: circ : d :: 22 : 7 ; circ = (3 + 1/7) ∙ d versus d = (⅓) ∙ circ because circ = 360° and d = 120p d : circ not expressible numerically
→ 2 2 1 2 Algor. (ii) circ A circ → (circ) → 7(circ) → ( ∕88) ∙ [7(circ) ] = A ps.-Hero No. 3 (17.3)
Algor. (iii) circ → A circ → circ + (circ/2) + (circ/4) = A ps.-Hero No. 4 (17.3bis)
Algor. (iv) d → A d → (d)2 → 11 ∙ (d2) → (1/14) ∙ [11d2] = A ps.-Hero No. 1 (17.4bis)
Second discussion of the ratio circ : d 8 Second quotation from Ptolemy’s Almagest: circ : d = 3 + ∕60 Minimal difference from the method, whence the use of the geometers’ method
Algor. (v) (d, circ) → A (d, circ) → d ∙ circ → (¼) ∙ (dcirc) = A ps.-Hero No. 2 (17.1)
Even though he modified the order of the subsections, there is no doubt that Komtino relied exclusively on codex S and no other manuscript of the pseudo- Heronian corpus. First of all, the astonishing algorithm (iii), which provides a correct result if and only if circ = 22; this algorithm appears in only one locus in the pseudo-Heronian corpus (Geometrica 17.3bis),73 which is not transmitted in the other branch of the manuscript tradition of this corpus (A and C). It is true that Vat. gr. 215 copied the three subproblems of Geometrica 17.4bis +17.3+17.3bis from S and it is true that several versions of the subproblem Geometrica 17.1 can be found in other collections of the Heronian corpus and thus in other manuscripts: Geometrica 17.18 (AC), Geometrica 21.5 (A), Diophanes 2 ( ). But S is the only manuscript with all four subproblems; it is the only one to illustrate the algorithm of Geometrica 17.1 with a circle (d=7, circ=22)—just as in Komtino—whereas the other three versions do it with a circle (d=14, circ=44). Finally, and above all, only “Euclid”’s Geometry and Komtino’s text describe the fifth algorithm as based on the sole diameter, whereas the method employed actually involves both the diameter and the perimeter:
73 Pseudo-Hero speaks of an alternative method; Komtino, of another example.
210 Tony Lévy and Bernard Vitrac
Another method for finding the area of a circle from its diameter ( ) [Komtino: And again, if you wish to find the area of Multiply the feet of the .[(מהקטר לבד) a circle from its diameter only diameter by the 22 feet of the perimeter, yielding 154 feet; take ¼ of this: 33½ feet: the area will be just as many feet.
However expressed, the relationship between d and circ is required. Komtino discusses it twice, in what he represents as a comparison of the statements by the geometers and by Ptolemy in his Almagest. The idea of this comparison, but not its formulation, may perhaps be borrowed from Bar Ḥiyya (§97). However:
(1) Bar Ḥiyya provides a quantitative comparison of the two methods and Komtino does not. where ,(אנשי המדידה) Bar Ḥiyya refers to astronomers and surveyors (2) Komtino has geometers and Ptolemy’s Almagest.
Note in passing that in the second quotation from Ptolemy, about the sexagesimal 8 74 approximation of π, Komtino gives the value of 3 plus ∕60. This value is too small: in Almagest VI.7, 512.25–513.5, Ptolemy proposes the approximation 3 8ʹ 30ʺ, which he explicitly compares to Archimedes’ limit values. Abraham Bar Ḥiyya, citing the astronomers (§97), offers the value of 3 + (8½ / 60), which “correctly” reports the statement in the Almagest. The “½” is missing in all five Komtino manuscripts we consulted on this point;75 this shared mistake confirms that all the surviving manuscripts derive from a common archetype. All in all, and contrary to what can be observed in other sections, in this section Komtino does not offer any arguments or proofs about the measurements of the circle, in particular not those given by Bar Ḥiyya, such as the superb “quadrature of the circle” at the end of §95.76 This may be why he twice mentions the “experts in geometry” and twice mentions Ptolemy, citing them in some way as “authorities.”
8 74 Silberberg’s ∕61 (“Ein handschriftliches hebräisch-mathematisches Werk II,” p. 224) must be 8 corrected; he failed to see that the reference was to ∕60 of a unit. Note that this error can already be found in Steinschneider’s description of the treatise, Mathematik bei den Juden, p. 198. 75 Here Pet. 343/4 is illegible. 76 See Tony Lévy, “Les débuts de la littérature mathématique hébraïque : la géométrie d’Abraham Bar Ḥiyya (XIe-XIIe siècle),” Micrologus 9 (2001): 35–64, on pp. 63–64.
211 Hero of Alexandria and Mordekhai Komtino
There are not as many Heronian borrowings in this section as in the previous ones, but nevertheless we see that Komtino did not feel bound to blindly follow a single source and then another, juxtaposing excerpts from earlier works. Even if the level of the mathematics is modest, he combines his two main sources according to the plan he set himself.
The Fourth Encounter: The Measurement of Regular Polygons The determination of the area of regular polygons, from the pentagon through the dodecagon, when the length of the side is given, is a specifically Heronian topic. It is dealt with five times in the Heronian and pseudo-Heronian corpus; four of these occurrences appear in manuscript S, and only there. They are treated in four different formats: a series of (eight) problems with demonstrations,77 a series of (eight) problems phrased as algorithms,78 formulas that state equivalences in area,79 and procedures given without numerical examples.80 The reason for this redundancy is that the compilers of the pseudo-Heronian problem collections wanted to include, as algorithms, what the Metrica treated from a double perspective, both demonstrative and algorithmic.81 We should note further that,
77 Metrica I.18–25 (S, fols. 76v–81r). Each polygon is the object of a unique measurement, itself justified by a proof, with a side of 10 units for all eight polygons. 78 Diophanes, §§10–17 (S, fols. 19r–20v, under the heading “Method of Polygons”). In
Diophanes 10.1, 12–17, we have CN = 10, as in Metrica I; the algorithms are the same as
those in Metrica. In Diophanes 11.1–2 (hexagon), we have C6 = 30 and the two algorithms proposed are not the same as those in Metrica 1.19; for the decagon, the other algorithm is said to be more accurate ( ) than the first one (just as in Geometrica 22.12). In each of the eight cases, the measurements are in feet in S and in units in Par. gr. 2448. This approach is also found in the other branch of the pseudo-Heronian corpus: Geometrica 21.14–23 (A, fols. 127r–128r; C, fols. 59r–60r). Geometrica 21.15 and
21.18–23 have CN = 10 (in feet), just as in Metrica I; Geometrica 21.16–17 has C6 = 30 feet,
as in Diophanes 11.1–2, Geometrica 21.14 has C5 = 35 feet. 79 Diophanes, §19 (S, fol. 22r). This section repeats the (8 + 2 =) 10 algorithms previously indicated in a list of equalities, without numerical examples, in a format inspired by the second proposition of Archimedes’ Measurement of the Circle (11 squares = 14 circles, which begins the list). 80 Geometrica 22.7–14 (S, fols. 61v–62r; A, fol. 129v). 81 See Héron d’Alexandrie, Metrica, pp. 450–455, 461–462, 484–488.
212 Tony Lévy and Bernard Vitrac perhaps unexpectedly,82 very few works on metrological geometry deal with these questions. There are isolated examples, especially for the pentagon and hexagon, in the Arabic literature on the subject,83 but not the full list. The fact is that once a table of chords has been established for the circle, formulas of this kind are of scant utility. Hence it is quite significant that Komtino, after a general comment about the possibility of measuring polygons by means of triangulation,84 includes such a series of problems in which he measures the area of eight regular polygons, from the pentagon through the dodecagon. As we shall see, he actually does this twice. The first presentation (NY, fols. 48a–49b) consists of a series of eight procedures with numerical examples. The three possible sources for this in the Heronian and pseudo-Heronian corpus are Metrica I.18–25 (S), Diophanes 10–17 (S + Par. gr. 2448), and Geometrica 21.14–23 (AC). We should add that except for the hexagon and the alternative algorithm proposed for the decagon, said to be more accurate, the procedures and numerical values for the other seven polygons are the same in all three versions.85 The only differences are in the treatment of the hexagon. In Komtino’s first presentation, the source is not the Metrica, which presents a proof (see below) that employs a totally different algorithm for measuring this polygon. This point also makes it possible to distinguish Diophanes from Geometrica 21, which presents two distinct (but mathematically equivalent) variants of the measurement algorithm:
1 2 • Diophanes 11: c6 → (⅓+ ∕10) ∙ [6(c6) ] = A. 1 2 • Geom. 21.16: c6 → 6 ∙ [(⅓ + ∕10 ∙ (c6) ] = A.
82 There is no comparable treatment of regular polygons in the work of Bar Ḥiyya, nor in Fibonacci’s Practica geometriae. We can add that it is also not found in Ibn ʿAbdūn, Abū Bakr, Ibn al-Haytham, or Joannes Pediasimus’ Geometry. 2 2 83 For example the algorithm C6 → √{(6 ½ ¼)[(C6) ] } = S (C6 designates the side of the hexagon) in Thābit ibn Qurra’s On the Measurement of Plane and Solid Figures (R. Rashed, ed., Thābit ibn Qurra. Science and Philosophy in Ninth-Century Baghdad [Berlin: Walter de Gruyter, 2009], p. 188]. 84 This can also be found in Bar Ḥiyya, §126. 85 With the minor exception of the alternative algorithm proposed for the pentagon in Geometrica 21.14.
213 Hero of Alexandria and Mordekhai Komtino
Komtino uses the variant found in Diophanes. The two authors’ use of an alternative algorithm (said to be “more accurate”) for the decagon points in the same direction, because it is not found in Geometrica 21.14–23 (AC). This comparison leaves no doubt: Komtino’s source, including for the hexagon, is each of the first parts86 of Diophanes’ problems, with two small changes:
• Komtino leaves out Diophanes’ alternative algorithm for the hexagon, which is merely a variant way of expressing fractions, whereas that for the decagon provides more interesting mathematical information. This difference suffices to explain why Komtino leaves it out. • More interesting are our author’s additions to or “hybridization” of Diophanes’ algorithms. Consider the decagon, for instance:
Diophanes 15 (MS S, fol. 20r–20v) Komtino (NY, fol. 49a–49b)
(1) Let be given an equilateral and equiangular And if you want to measure the figure with decagon, with each side 10 feet; find its area. 10 sides and 10 angles, proceed as follows: I proceed as follows.
(2) —— Multiply the side by itself and the result, multiply it by 15 and take half of it, which is the measurement of the area.
(3) —— Example: We have a figure ABGDHWZḤṬI, each of whose sides is 10 cubits.
(4) [Multiply] the 10 by itself, yielding 100; We multiply 10 by itself: 100 this, multiplied by 15, yields 1500. Take the We multiply by 15, which makes 1500 half, yielding 750. And half is 750
(5) So much will be the area of the decagon, And that is the area of the figure. 750 feet.
86 For a reason that is not clear, in three of the eight cases (pentagon, octagon, and nonagon) after the measurement of the area of the polygon, Diophanes introduces a second question dealing with the diameter of the circle circumscribed around the polygon. Komtino did duplicate these three extra questions. They could be helpful if one wishes to measure a pyramid whose base is such a polygon, knowing the inclination (i.e. slant edge), in order to calculate the height. But nowhere is there any trace of a pyramid with a nine-sided base. What is striking in the treatise by Diophanes is its total lack of system.
214 Tony Lévy and Bernard Vitrac
Diophanes 15 (MS S, fol. 20r–20v) Komtino (NY, fol. 49a–49b)
(6) Alternatively, again, And in a different way:
(7) [Multiply] 10 by itself, yielding 100. We multiply 10 by itself: 100. [Multiply] that by 38, yielding 3800. We multiply by 38 and this makes 3800. From that, always, [take] one-fifth, yielding We take one-fifth of this, which is 760. 760.
—— And this is the area of the figure.
This method is more precise […] And this is a more precise calculation.
Komtino’s version is not simply a translation: where Diophanes merely offers an algorithm based on a specific numerical example (1 and 4), Komtino first explains the procedure in general terms (2) and then gives the example of a figure with named vertices (3), as found in the text of demonstrative geometry on which he bases the algorithm (4). With regard to the procedure, formulated independently of the numerical data and thus with universal validity,87 Komtino may have developed it himself or found it elsewhere in S, and in particular in the short set of formulas attributed to Euclid.
(pseudo-)Euclid, Euthymetrika (Geom. 22.12), MS S, fol. 62r
(1) Find the area of a decagon
(2) [Multiply] the side by itself; that by 15; half of which is the area.
(3) Alternatively, again,
(4) [Multiply] the side by itself; that by 38, of which a fifth.
(5) This is the area.
(6) This [method] is more precise.
87 When only algorithms are presented, it is necessary to multiply the numerical examples (1) in order demonstrate the generality of the procedure which, with very rare exceptions, is not valid for specific data only (but see above, Algorithm 3 of the section on the measurement of the circle) and (2) so that readers can know which numerical values are the givens or partial results and which are the universal constants of the algorithm in question; in this example (rather simple), 15, ½.
215 Hero of Alexandria and Mordekhai Komtino
Komtino’s use of Diophanes can be explained by the strange character of the corresponding section in Abraham Bar Ḥiyya, whose heading, “Figures with More than Four Sides,” does not correspond to its content. It includes a triangle and a square circumscribed around a circle (§§125–125a); then, after dealing with the pentagon—the only example of a regular polygon (§§125b–126)—it switches to problems of geodesy, such as measuring the height of a mountain or the area of a skew surfaced field. Komtino may not have been satisfied with this; so, starting with the pentagon, he follows Diophanes, because he employs the same algorithm, which Bar Ḥiyya does not, being content with generalities (at least in the current state of the text as established and edited by Guttmann). At the end of the example of the dodecagon, Komtino adds a remark that is not found in Diophanes (but see Bar Ḥiyya, §126):
And the other figures with many sides, in which the sides and angles are not equal, are measured by the triangles into which they can be divided.
But that is not the end of § 1.6. After the presentation of the procedures, the text continues (NY, fol. 49b–50a):
Mordekhai said: I have acquainted you with the figures with multiple angles, from the pentagon through the dodecagon, easily and without difficulty, and without having brought a proof, so that the student will not be confused, and I have provided the method to be followed. Before going further, however, I will offer the proof for each of the polygons and I will begin with the pentagon, as I did in a practical manner. And before beginning to supply the proof, I will begin by mentioning two triangles that will be of use for what we wish to establish.
What follows (NY, fols. 50a–54a) is in fact a version of the Metrica, Propositions I.17–25 (S, fols. 77v–81r), with the same lemmata and diagrams. The transition just quoted may seem somewhat misleading: it suggests that the first two propositions of this section are lemmata on triangles and that the actual sequence of proofs begins with the pentagon. In fact, although the second is indeed a lemma—which leads into the proof of Metrica I.18, on the pentagon—the first is a faithful translation of Metrica I.17 on the measurement of an equilateral triangle. Komtino seems to consider it to be a lemma, so as to produce a continuous series of propositions—the square is not measured as part of the series of regular
216 Tony Lévy and Bernard Vitrac polygons—and most of all to maintain the correspondence with the previous sequence, taken from Diophanes. It is possible that he perceived it as preparation for the measurement of the regular hexagon (itself thought of as six equilateral triangles) in Metrica I.19. It bears note that the new material he inserts is probably one of the most idiosyncratic Heronian propositions. To the best of our knowledge, no author, ancient or medieval (aside from Hero’s own allusion in Metrica I.19), ever referred to this proposition of the Metrica and the original algorithm it presents: 2 4 3 4 3 4 [c → c → c → ( ∕16) ∙ c →√{( ∕16) ∙ c }=A]. The text runs as follows:88
(i) To begin with, let there be given an equilateral triangle with sides of 10 units. Let it be .
(ii) Let a straight line be drawn perpendicular to .
(iii) And because —which is the same as —is twice , the [square] on is thus 4 times that on ; so that the one on is triple that on ; or that on is four times that on : that on is consequently four-thirds of that on ; that on relative to that on thus has a ratio of 4 to 3.
(iv) And let all of them [be multiplied] by that on —whence, also that on [multiplied] by itself and that on [multiplied] by that on
(v) 89 The dunamodunamis on thus has a ratio to the square on by that on , which is 4 relative to 3, that is, of 16 relative to 12;
(vi) But the [square] on by that on is the [rectangle contained] by , by itself— that is, two triangles by themselves;
(vii) 90 The dunamodunamis on relative to two triangles by themselves thus has a ratio of 16 to 12;
88 Héron d’Alexandrie, Metrica, pp. 190–191. 89 S 90 S sed - suppl. in mg. m. 1
217 Hero of Alexandria and Mordekhai Komtino
(viii) Now two triangles [multiplied] by themselves are four times a single triangle [multiplied] by itself;
(ix) 91 The dunamodunamis on relative to a single triangle [multiplied] by itself thus has a ratio of 16 relative to 3;
(x) And the dunamodunamis on is given—because is also.
(xi) The area of the triangle by itself ( ) is thus given; so that the triangle itself is given.
(xii) And in consequence of the analysis, it can be synthesized this way.
(xiii) [Multiply] 10 by itself; the result is 100; [multiply] that by itself: the result is 10,000; take 3 ∕16 of this: the result is 1875; take one side of this;
(xiv) And because the side cannot be expressed [i.e., is irrational], approximate it with a difference, as we have learned;
(xv) And the area is 43⅓.
This section of the Metrica is also that with the most geometrically bizarre expressions: the product of one square by another (iv, v, vi) or by itself (iv), the product of one (or two) triangles by it/themselves (vi, vii, viii, ix), the product of a rectangle by itself (vi), the area of a triangle by itself (xi), and four occurrences of the word dunamodunamis (v, vii, ix, x), generally described as a term borrowed from Diophantine arithmetic, but which is anticipated by Hero and slightly modified, because it is the dunamodunamis of a straight line (in fact of its length). What is more, as indicated by our notes 89-91, the text of S has mistakes in three of the four occurrences of the phrase (the dunamodunamis of line ); these errors are probably the result of the copyist Ephrem’s misexpansion of unfamiliar abbreviations found in his copytext; only the fourth instance is written correctly. These “variants” must have perplexed Komtino. Here is his translation of the passage (according to NY, fol. 50a):
91 Schöne: S
218 Tony Lévy and Bernard Vitrac
Thus the power of square of BG relative to אם כן כח מרבע בג אצל מרבע בג על מרבע (v) ,square of [B]G [multiplied] by square of AD אד יחסו כיחס ד אל ג, שהוא כיחס יו אל יב. its ratio is that of 4 to 3, which is the ratio of 16 to 12;
And the square of BG [multiplied] by the ומרבע בג על מרבע אד הוא ככפל השטח (vi) square of AD, is the same as the product of הנצב הזויות אשר יקיפו בו אדבג על עצמו, the rectangular surface enclosed by AD, BG כלומר שני משלשים על עצמם. [multiplied] by itself, that is, two triangles [multiplied] by themselves;
If so, the power of its power of square of BG אם כן כח כחו של מרבע בג על כפל שני (vii) ,to the product of two triangles by themselves משולשים על עצמם, יחסם כיחס יו אל יב, their ratio is the same as the ratio of 16 to 12;
And the product of the two triangles by וכפל שני משולשים על עצמם הוא ארבעה (viii) themselves is 4 times the product of a triangle כפלי משלש אחד על עצמו, by itself;
If so, the power of its power of square of BG to אם כן כח כחו של מרבע בג אצל כפל משלש (ix) the product of a triangle by itself, its ratio is the אחד על עצמו, יחסו כיחס יו אל ג. same as the ratio of 16 to 3.
And the power of its power of square of BG is וכח כחו של מרבע בג ידוע, (x) known
If so, also known is the area of the triangle by אם כן גם תשברת המשלש על עצמו ידוע (xi) .itself, and from that the triangle itself is known ולכן גם המשלש עצמו ידוע.
:The procedure is as follows ואופן המעשה כן (xii)
(xiii) [Multiply] 10 by itself: the result is 100; [multiply] that by itself—the result is 10,000; from 1 92 this take one-third of half [of 10,000] plus ∕24; the result is 1875; from this take one side
(xiv) And because it does not have a root that can be expressed [numerically, i.e., it is irrational], I take its root by approximation;
1 1 1 (xv) And the area is 43 ⅓ ∕38 ∕40 ∕41.
Komtino rendered the first occurrence of dunamodunamis (in fact, ) as “the power of the square of BG” and the other three as “the power of its power,”93—which, taken literally, would seem to refer to an
92 All the available manuscripts have the same reading, 1/24, which clearly is a mistake. To make sense, the text should read 1/16 here. The calculation must be understood as follows: take one third of (1/2 + 1/16) of 10000; the result is 1875. 93 With this awkward and pleonastic translation we are trying to represent the difference in .כח כחו של the Hebrew of the two occurrences of “power,” here
219 Hero of Alexandria and Mordekhai Komtino eight-dimensional entity. This variant of Komtino, as found in NY, is in some sense faithful to the mistakes in S; it clearly confused the copyists of the later manuscripts.94 After the end of the Heronian analysis based on a given (known) magnitude, Komtino transcribed Hero’s numerical procedure. But instead of offering it as a (numerical) synthesis ( ), he presents it as a practical method. In his versions of Metrica I.18 and I.20–25, Komtino quite simply drops the Heronian numerical synthesis, because the procedure, based on Diophanes, has already been presented using the same numerical example. Then he applies to the entire set the same procedure followed in Metrica I.12–14 and 16, as we have already seen; that is, first he lists the algorithms (here taken in toto from Diophanes, with some variations), and then provides a proof for each of them, taken from the Metrica.
The two ostensible exceptions are not really exceptions: • The practical method provided here and retained at the end of the version of the Metrica I.17. This is legitimate, because the equilateral triangle is not part of the sequence borrowed from Diophanes. • It is also retained for the regular hexagon, at the end of the version of Metrica I.19 (thus without inversion). Here Komtino remarks, quite correctly, that it is different (both structure and data) than that presented previously (based on Diophanes): And because the method for doing this proceeds by different aspects than those we mentioned at the start, I will write this: multiply 10 by itself and you have 100; 100 by itself and you have 10,000. Take a quarter of this, which is 2000 and 500. We multiply it by 27; this yields 67,000 and 500. Take the root, which is 259. And this is the area of the hexagon.
Komtino’s attention to his Greek model was very close; we will provide one last amusing example. In Metrica I.25, Hero measures a regular dodecagon of side 10. The approximative algorithm is based on elementary measurements of angles and 7 the classic approximation that the square root of 3 ≈ ∕4.
94 See below, Appendix C.
220 Tony Lévy and Bernard Vitrac A B N On the diagram opposite, triangle is equilateral.95 M 2 2 G Hence equals 3 ∙ And : ≈ 7:4. By construction and Euclid’s Elements X L D I.6: = And thus
K : :: 8:15 (= 8+7) E P But the eta for “8” is missing in codex
Q S, and Ephrem left a blank space there Z O H (see Fig. 3):
Blank space where there should have been an (= 8) no blank space
Figure 3: MS S, fol. 80v lines 22–23 apud Bruins (Brill, 1964, supra note 31), p. 156
The same ratio reappears in the next line (in modern transcription, 2 : × :: 8 : 15); once again the numeral 8 is missing, but this time there is no blank where it should be. In his translation, Komtino adhered to the layout of his model: in NY, the first omission of the digit is marked by a blank space, as in S; the second time, again as in S, there is no blank space. However, a siglum (three superimposed dots) refers to a marginal note about the lacuna: “[read what is] .(see Fig. 4) ”8 :( חסר) missing
1 95 Because = ⅓ of a right angle (30°) and = = ∕6 of a right angle (15°) by construction; whence = ⅓ of a right angle (30°) and = ⅔ of a right angle (60°).
221 Hero of Alexandria and Mordekhai Komtino
Marginal note: Reference to the Blank space where 8 This is the note to which marginal note should have been written
refer the three dots above Marginal note: Reference to the marginal note Blank space where 8 should theIt lineseems in to the me: text. 8 missing It have been written means: “It seems to me Figure 4: MS NY, fol. 54a, lines 5–14 Figure 4: MS NY, fol. 54a, lines 5–14 [that] 8 is missing.” Courtesy of The LibraryCourtesy of the of Jewish The Library Theological of the Seminary Jewish Theological Seminary
In St.In StPetersburg. Petersburg 343/4343/4,, B, B,P, andP, andL, the L space, the hasspace vanished has andvanished we find, and twice we, ―is find, twice, “is likelike thethe ratio ratio to 15 to‖ (15”sic). (sic). InIn codex codex Pet. Pet. (see (see Fig. 6),Fig. the 5), missing the missing8 was inserted, 8 was no inserted, doubt later, no because doubt the later, because characters are spaced out: ―is like the ratio of 8 to 15.‖ the characters are spaced out: “is like the ratio of 8 to 15.”
TwoTwo occurrences occurrences (ח) 8 (ח) of theof digitthe 8digit wherewhere the text the is spacedtext is out spaced out
Figure 5:Figure MS Pet. 5,: fol.MS 32a Pet., fol. 32a
For all that the measurement of regular polygons might seem an unfashionable topic at the end of the fifteenth century, it nevertheless provides abundant indication of how Komtino employed his sources. He changed them when he was not satisfied with 222 35
Tony Lévy and Bernard Vitrac
For all that the measurement of regular polygons might seem an unfashionable topic at the end of the fifteenth century, it nevertheless provides abundant indication of how Komtino employed his sources. He changed them when he was not satisfied with them, read the Greek text with care, and adapted its dual approach of both proofs and algorithms/procedures when he took over two passages from the Metrica. His procedure is simpler in the last two encounters, which deal with the measurement of solids: here all the borrowings come from the pseudo-Heronian collections of problems.
Fifth Encounter: The Measurement of Pyramids Komtino begins the section on the measurement of solids (as Bar Ḥiyya did in §152) with the definition of a solid. Next he inserts a list of figures, illustrated by various sketches, which on first reading does not appear to have any system to it: pyramids with triangular, quadrangular, circular (i.e., a cone), and polygonal bases; prisms with a triangular base; a truncated pyramid with a square base; a sphere. Thus he is not motivated either by general considerations that guided Hero in the measurement of volumes at the start of the second book of the Metrica, notably the enunciation of a specific form of what we know as Cavalieri’s Principle, or by the brief and rather original classification with which Bar Ḥiyya began his chapter of the measurement of solid bodies. The latter effectively divides solids into three classes, different genera and species (§§153–155), as follows:
Class 1: Right solids with the base and the face opposite it parallel and equal. Genus 1.1: The base, opposite face, and all other faces are quadrangular. Examples (§§156–160): cube, “long” prism, parallelepiped;96 solids whose base is a rhombus, rhomboid, trapezium. Genus 1.2: Both base and opposite face are triangles or polygons; the other faces are quadrangular (§161). Example, a prism with a triangular base or a partial prism. Genus 1.3: The base and opposite face are circular, the “lengths” are rectilinear. Example (§162): cylinder.
96 This first distinction considers three cases of rectangular parallelepipeds with sides (a, b, c) such that (a = b = c), (a = b ≠ c), and (a ≠ b ≠ c).
223 Hero of Alexandria and Mordekhai Komtino
Class 2: Decreasing solids Genus 1: The top is a point. Examples (§§163–164): pyramid, cone. Distinction based on the height, the lateral edge, the height of the triangular faces (§165); this leads to the classification of pyramids into four species (§166) on the basis of two criteria: Equality/inequality of the sides of the base Equality/inequality of all the “straight lines rising from the base הקוים היוצאים מזוית תושבתם ...) ”vertices to the top of the pyramid lit. “from the base angles to the head of the pyramid”).97 ; אל ראש Species 3 and 4 (in which the inclinations are unequal) are called ,or as an illustration (הרחבת הלשון) pyramids” only by extension“ Bar Ḥiyya tells us; consequently, he does not discuss them further. In §§167–170, Bar Ḥiyya explains how to find the height when this is not a given; we will return to this. Genus 2: The face opposite the base is the same figure as the base, but smaller. Examples (§§171–177): truncated pyramid and cone. Class 3: Fully circular solids (§§178–180). Examples: spheres and their segments.
This impressive classification is inspired in part by an algorithmic logic. Bar Ḥiyya notes that all the solids of Class 1 are measured in the same way, using the formula V=Bh (where V is the volume, B is the area of the base, and h the height). Similarly, those of the first genus of Class 2 are measured by the same formula V=B(⅓ h). Still more remarkable is the case of the solids of the second genus of Class 2, which are measured in one of two ways: either by completion—we imagine a pyramid (or cone), and the volume of the tapering solid is the difference between two solids of the first genus (§172); or by an alternative procedure, first presented for truncated pyramids with a square (§173) or triangular (§176) base and for a truncated cone (§174), and then generalized (§177):
In this manner one calculates the volume of all truncated pyramids, with a base that is a pentagon or any other polygon: by adding the areas
97 Stated another way, the oblique edges of the pyramid, straight lines that the pseudo- Heronian corpus refers to as inclinations ( ).
224 Tony Lévy and Bernard Vitrac
of the two bases and the square root [of the product] of their areas, and multiplying the sum by one-third of the height, the product is the volume,
Which can be written as follows: V= [B + S + √(B ∙ S)] ∙ [(⅓) ∙ h], where B and S are the areas of the base and opposite face, respectively.
***
If we reread the introduction to Komtino’s §3 in light of the above, we see that he does not present the principles or follow the order (he does not even mention Class 1), but selects some examples and reorders them according to other criteria. This is followed by the measurement of a cube with side 10, and then of its diagonal (marked by use of the cardinal points northeast/southwest rather than by the letters of a standard geometrical diagram], and then of a rectangular parallelepiped with sides (10, 15, 20) and its diagonal. Although the example of a cube and its diagonal can be found in Bar Ḥiyya ,אמה מוגשמת) and 159), as can the short couplet defining the solid cubit 156§§) NY, fol.60b, l.7)98, this does not seem to be the case for the choice of the parallelepiped: Bar Ḥiyya (§§157–158) uses (4, 4, 10) and then (6, 7, 8). Nor is Komtino’s example (10, 10, 20) found in the Heronian and pseudo-Heronian corpus.99 Komtino’s presentation then copies Bar Ḥiyya’s classification, without naming the genera and species, but only listing the solids whose volume is determined by the same method:
98 Bar Ḥiyya, in §156a, mentions other metrological equivalences valid for Spain that Komtino discarded. 99 Another troubling difference: our two authors calculate the diagonals; for a cube of side 10, Bar Ḥiyya is content to say that it is equal to √300, whereas Komtino (Silberberg, “Ein handschriftliches hebräisch-mathematisches Werk II,” p. 228) gives the value 17 1 ¼ ∕15 (which approximates √3 = 1.73166…). Either Komtino calculated this excellent approximation (not attested in the Heronian and pseudo-Heronian corpus) for himself or borrowed it from another source, or the text edited by Guttmann and translated by Millàs i Vallicrosa is a less complete version of the Ḥibbur ha-meshiḥa we-ha-tishboret than that used by Komtino. See also n. 103.
225 Hero of Alexandria and Mordekhai Komtino
• V = B ∙ h (Class 1) • V = B ∙ [(⅓) ∙ h] (Class 2, Genus 1) • V = [B + S + √(B ∙ S)] ∙ [(⅓) ∙ h], illustrated by the same three truncated solids (NY, fols. 62a–63a; B, fols. 80b–82a; P, fol. 60a–b):
(a) Truncated pyramid with a square base of dimensions (C4 = 6, c4 = 4, h = 10)
(b) Truncated cone with dimensions (d1 = 4, d2 = 2, h = 12)
(c) Truncated pyramid with an equilateral triangle base (C3 = 8, c3 = 4, h = 6).100
The rule is formulated in general terms (NY, fol. 63b) in a fashion very similar to what appears in Bar Ḥiyya:
And by this method, you will know the measurement of any figure truncated at its top whose base is a pentagon or any other figure. To השני ראשים proceed you will know the area of the two opposite faces (sic literally “the two heads”101) and to these you add the products of the root ;by the root of the area of the base (הראש) of the area of the opposite face add all these together and multiply by one-third the height and you have the volume of this figure.
Komtino was also inspired by Bar Ḥiyya’s remarks on determining the height of different types of pyramids (§§165, 167a–168, 170) and the measurement of barrel-shaped solids (§175, which can be assimilated to the sum of two conical frustra). This is a sequence that Komtino borrows from his predecessor and then modifies. Note that Bar Ḥiyya’s presentation of the measurement of solids is very compact, theoretical, and not always “practical.” It is important to underscore that an entire category of solids can be measured by means of the same formula, such as V=B∙[(⅓)∙h]; but how does one measure the magnitudes of B and/or h if they are not among the givens of the problem?
100 CN (or d1) designates the side (or diameter) of the base, cN (or d2) that of the top, and h the height. 101 Here we would expect “the area of the base and the area of the top,” but Komtino, following Bar Ḥiyya, wants to highlight the symmetrical role of the two magnitudes in the rule by using the same term for both rather than the standard couple top/base.
226 Tony Lévy and Bernard Vitrac
Bar Ḥiyya provides several general and vague indications that at first glance may seem to be somewhat incoherent:
• In §166, introduces his classification into four species by referring to “straight lines that proceed from the angles of the base to the top (of the pyramid)”; in other words, here we are dealing with the oblique edges of a solid. • In §§167a–168, Bar Ḥiyya presents a single procedure that can be applied to figures whose base is an equilateral triangle or other regular polygon, a procedure that introduces, without making it explicit, the radius of the circle inscribed in the base (given by r = 2A/p, where A is the area of the base and p its perimeter). Next, its square is subtracted from the square of the “straight line that runs from the base to the top” to obtain the height of the pyramid. For this expression to actually yield the square of the height, the line in question must be the apothem (i.e. the height of the slanted faces) and not the inclination (i.e. oblique edges). • In §170, to illustrate the case of a pyramid with an irregular—in fact, oblong—base, he explains that in order to find (the square) of the height, one must subtract from the square of “the straight line that runs from the vertices of the base to the top” that of the semi-diagonal of the rectangle (which is in fact the radius of the circle circumscribed around the base).
The reasons for these variations is that if the given magnitude is the inclination (or the apothem), one must determine the radius of the circle that circumscribes102 (or is inscribed in) the base in order to calculate the height of the pyramid by the method of the difference of the squares. This does not involve any real computational difficulty because, if one knows the side(s) of the base, a simple application of the Pythagorean theorem makes it possible to determine the
102 We should observe that for right pyramids with a square base, there is no need to make explicit use of these radii: the radius of the inscribed circle is equal to half the side, that of the circumscribed circle is equal to the semidiagonal of the square base. Similarly, for a pyramid with a regular hexagon as base, the radius of the circumscribed circle is equal to the side of the hexagon. It is only in the case of pyramids with a pentagonal or octagonal base (for which Komtino borrowed examples from the pseudo-Heronian corpus; see below) that the circumscribed circle is required.
227 Hero of Alexandria and Mordekhai Komtino apothem from the inclination (and vice versa). It is quite likely that Bar Ḥiyya considered this relationship to be self-evident. Nevertheless:
• The Hebrew text103 seems to switch back and forth, depending on whether it is defining the types of pyramids or presenting the various procedures. • Bar Ḥiyya does not offer any numerical illustration of the procedures, which, whether with the inclination or the apothem as given, would eliminate the ambiguities. *** Komtino, following Bar Ḥiyya, also raises the question of how to determine the height (NY, fol. 61b):
As for knowing how to find the height of a figure, I will tell you: Know that the height is a perpendicular; it runs from the base of the figure and proceeds at right angles towards the top.
After this, he found a type of didactic problem already experienced by the ancients. Hero, in the Metrica, and Bar Ḥiyya after him, insisted on the general applicability of their procedures, such as that for the measurement of pyramids of any base, without offering details. As a result, the authors of the pseudo- Heronian corpus assembled collections of problems for pyramids in which what is given is not the height, but the oblique edge(s) called inclination(s) ( ; .104 The first requirement was then to find the height (which depends on the nature of the base).
103 The Latin version by Plato of Tivoli (Der Liber embadorum des Savasorda, in Curtze, Urkunden zur Geschichte der Mathematik, pp. 166, 168) is more forthright in this variation; when it classifies pyramids, it has “lineae, quae ab angulari basis […] ad extremum piramidis punctum diriguntur” (i.e. inclinations); in its algorithms, “lineae, quae a dimidio cuiuslibet lateris eiusdem figurae usque ad extremum punctum piramidis protrahitur” (i.e., apothems). This variation is another indication (see above, n. 99; below, nn. 142-144) in favor of Steinschneider’s belief that there were two different recensions of Bar Ḥiyya’s text. See Lévy “Les débuts … : la géométrie d’Abraham Bar Ḥiyya (XIe-XIIe siècle),” pp. 45–47. 104 As stated, in a pyramid with a regular polygon as base (including an equilateral triangle and square) these are the straight lines (of equal length) that run between the top of the
228 Tony Lévy and Bernard Vitrac
There is no doubt that it was the variations in Bar Ḥiyya’s text, his sometimes implicit formulas, and the lack of examples that led Komtino to switch to a different source. Hence it is not surprising that his text proceeds by what can be described, following Silberberg, as a series of exercises on this theme (NY, fols. 63a–67b; Pet., fols. 36a–38b; B, fols. 81b–89a; P, fols. 61a–63b), introduced by:
Now I will give you an example of many figures so that you can learn how to calculate their volume yourself (NY, fol. 63b).
What follows is simply a translation of a long extract taken from the collection of problems in Stereometrica II.55–68, transmitted in codex S (fols. 55r–61r) under the heading (“measurement of pyramids”), arranged as follows:105
Stereom. II.55 = Silberberg, pp. 231–232, No. 1: Pyramid with square
base (C4 = 24, Kl = 18).
Stereom. II.56: Pyramid with square base (C4 = 10, Kl = 13 ½).
Stereom. II.57 = Silberberg, p. 232, No. 2: Pyramid with square base (C4 = 12, Kl = 36). Find the height and the volume. Stereom. II.58 = Silberberg, p. 232, No. 3: Truncated pyramid with
square base (C4 = 10, c4 = 2, Kl = 9). Find the volume.
Stereom. II.59.1–4: Truncated pyramid with square base (C4 = 16, c4 = 6, Kl = 40). Find the volume (calculated by average). Stereom. II.59.5: Truncated pyramid with an oblong base (sides: 16, 12). Calculation not made. Stereom. II.60.1–2 = Silberberg, p. 232, No. 4: Pyramid with right- triangle base (h = 25, sides of the right angle: 4, 5). Find the volume. Stereom. II.60.3–5 = Silberberg, p. 232, No. 5: Pyramid with isosceles triangle base [Base = (12, 12, 8), Kl = 25]. Find the volume. Stereom. II.60.6: Pyramid with obtuse-angle or acute-angle triangular base. Treatment by analogy.
pyramid and a vertex of the base (the “inclination”). To measure pyramids whose base is an isosceles triangle or rectangle (see below, Stereometrica II.60.3–5 and II.61), speaking of the inclination (in the singular) assumes that the top is located directly above the center of the circle circumscribed around the base, so that it is equidistant from all vertices of that base. 105 Kl (for ) designates the inclination of the pyramid or truncated pyramid.
229 Hero of Alexandria and Mordekhai Komtino
Stereom. II.61 = Silberberg, p. 232, No. 6: Pyramid with right-triangle base [Base = triangle (6, 8, 10), Kl = 13]. Find the height and the volume. Stereom. II.62 = Silberberg, p. 233, No. 7: Pyramid with equilateral
triangle base (C3 = 30, Kl = 20). Find the volume. Stereom. II.63 = Silberberg, p. 233, No. 8: Pyramid with pentagonal base
(C5 = 12, Kl = 35). Find the height and the volume. Alternative computation (incorrect) of the radius of the circle circumscribed around the base, from Eucl. XIII.10. Stereom. II.64 = Silberberg, p. 233, No. 9: Pyramid with hexagonal base
(C6 = 12, Kl = 35). Find the height and the volume. Stereom. II.65 = Silberberg, p. 233, No. 10: Pyramid with octagonal base
(C8 = 10, Kl = 15). Find the height and the volume. Stereom. II.66 = Silberberg, p. 234, No. 11: Fluted pyramid with
equilateral triangle base (C3 = 10, Depth of the grooves = 2, Kl = 20). Find the radius of the circumscribed circle, the height, and the volume (computed by difference). Stereom. II.67: Not a problem, but only a quotation from Euclid’s El. XII.7, with deduction of a general rule for calculating the volume of a pyramid. Stereom. II.68 = Silberberg, p. 234, No. 12: Bomiskos [h = 50, base = rectangle (24, 16), opposite face = rectangle (12, 8)]. Find the volume (calculated using the algorithm in Metr. II.8, but with different numerical data).
Here Komtino translated Stereometrica II.55, 57, 58, 60–68,106 but omitted II.56 and 59. In one sense, 55–57 pose the “same” question; Komtino retained only the two whose numerical data are in some fashion correlated: taking half of the side of the base and doubling the edge. What is more, II.55 and 56 use precisely the same algorithm, slightly different from that employed in II.57: the first calculates the square of the semi-diagonal directly from the square of the base [(d/2)2 = 2 (½)∙(C4) ], whereas the latter first computes the diagonal of that square [d = 2 2 √((C4) + (C4) )], then half of it, and then the square of the half. As for problems 58 and 59.1–4, both of them involve a truncated pyramid with a square base, but Komtino kept only the one with the correct algorithm. Stereometrica II.59, whose terminology is in any case somewhat unusual,
106 The loss of one folio of NY (originally fol. 64) has taken with it most of problem II.57 (only the first two lines, at the bottom of fol. 63b, survive), the start of problem II.60 (the current fol. 64a resumes with §60.2), as well as problems 57 and 58.
230 Tony Lévy and Bernard Vitrac employs the average of the areas of the base and opposite face, which strictly speaking is incorrect or at best approximative. All the problems for measuring a pyramid with a square or regular polygonal base that Komtino drew from S take the base and inclination as the givens; hence they must first calculate the height by determining, explicitly or implicitly, the radius of the circumscribed circle (see above, the discussion of Bar Ḥiyya’s text, §§165–170). The computation is explicit in the case of a pyramid with a pentagonal base, implicit for those whose base is an equilateral triangle,107 square, or regular octagon. The circumscribed circles are indicated on the diagrams of Stereometrica II.63 = Problem 8 (pentagon) and II.65 = Problem 10 (octagon), both in the Greek codex S (fol. 50r, fol. 50v) and in the Hebrew manuscript NY (fol. 65b, fol. 66a). Several problems in this set include meta-mathematical assertions combined with passages from Euclid: Stereometrica II.57.3, II.60.2, II.63.5, and II.67, all of them translated by Komtino, display some discomfort with the algorithm employed to measure these different pyramids. By way of example, here is the end of Stereometrica II.57:
Pseudo-Hero108 Komtino (B, fol. 83a)109 But why a third? But why do we take a third?
Because every prismatic solid can be divided Because every solid prism110 can be divided into three pyramids with height equal to the into three pyramids of the same height and height of the prism, and with triangles as with a base that is a triangle. And we perform
107 This relies on the well known fact (see Euclid, Elements XIII.12) that the square of the radius of the circle circumscribed about an equilateral triangle (or ⅔ of its height) is ⅓ of the square of the side of the triangle. See Stereom. II.62 = Problem 7. 108 Stereom. II.57.3, fol. 55v; in Heronis Alexandrini opera quae supersunt omnia, vol. V: Heronis quae feruntur Stereometrica et De mensuris, ed. J. L. Heiberg (Leipzig: B. G. Teubner, 1914), 140.2–10. 109 Pet., fol. 36a, B, fol. 82b, P, fol. 61a. This problem was on the now lost original folio 64 of NY. We follow Pet. The diagram in P (in the window at the bottom left) is very similar to that in Pet.; that in B, like all those in this part of the text, were drawn freehand and are very sketchy. 110 Following the reading of B, which here, as in several other passages, undoubtedly preserves Komtino’s original text (we are still in the lacuna of NY). On the other hand, in Pet. (fol. 36a) and P (fol. 61a), the word “prismatic” has been omitted, so the statement is false as a general rule.
231 Hero of Alexandria and Mordekhai Komtino
bases. Now we proceed as if we had a solid the calculations with a solid parallelepiped parallelepiped and the solid parallelepiped is and a solid parallelepiped can be divided into equal to two prisms, but each of these prisms two prisms and each prism of the height of is three times the pyramid taken by itself ( the pyramid is three times the half of the given 111) the [prism] pyramid; and that is what has a square base. on the [base] half of [that] of the proposed And this has been demonstrated by Euclid in pyramids; because it has a square base. Euclid Book XII. demonstrated this in Book XII.
A particularly instructive case is that of exercises 4 and 5 (= Stereom. II.60.1–2 and 3–5, S, fol. 57rv) in Silberberg’s numbering, because they include not only assertions of this type but also several mathematical errors:
1. To measure a pyramid with a right triangle as the base, it is not necessary to find the inclinations, because the perpendicular is at right angles ( 112 ). Let the perpendicular be 25 feet, the first side of the right angle113 4 feet, the other 5 feet. Do this. [Multiply] 4 by 5: the result is 20, of which [take] half: the result is 10. [Multiply] this by the 25 of the perpendicular; the result is 250; of which 2 114 [take] one sixth: the result is 41 ∕3).
First mistake: after calculating the area of the base triangle, 10, multiplied by the height, 25, one must take one-third of it, and not one-sixth.115 This yields
111 Note the curious locution to designate the pyramid “according to [the prism] itself”; in other words, one that has the same base and height. Komtino understood this phrase correctly. 112 One would expect “given” ( ) rather than “having right angles” ( ). 113 The adjective “right” is missing in all three manuscripts we consulted where this portion survives (thus not NY). This makes Komtino’s translation somewhat difficult to understand. 114 With a regard to Komtino, here we follow the reading of B, whose text is coherent and similar to that in S, but mathematically false: in the algorithm, one must use ⅓. 115 In Pet. (fol. 36b) and P (fol. 61b), we read “take ⅓, which is 41 and ⅔,” which is only a partial correction (because ⅓ of 250 is 83⅓). In the first, however, there is a marginal note 1 (with a reference mark in the text): “one has to use ∕6.” There is no similar “correction” in P. It is too unfortunate that we do not have the reading of NY for this passage, so we could know whether, as we suppose, it is the reading of B (and of S) that an early reader corrected
232 Tony Lévy and Bernard Vitrac
116 2 83⅓ and not 41 ∕3. This mistake must have troubled some reader in ancient 1 times, who wondered why one should take ∕6:
2. Why do we do this? Every prism with a triangular base, which is half of a quadrangle, can be divided into three pyramids with triangles for bases, and similarly to the prism. [The text is corrupt. The sense may be “of similar height as the prism.”] Euclid proves this in Book XII. If, then, the prism is half of a quadrangle (sic: of a parallelepiped with quadrangular base?) and can be divided into three pyramids, it necessarily follows that the [volume] of the pyramid with a triangle for a base is one-sixth […]117 having a base. After performing the quadrature ( ) [of the base], 1 we then take ∕6.
The lacunae in this paragraph did not make things easy for Komtino the translator, who labored to faithfully restore the sense of the Greek text. In addition to the latter’s incoherencies, we may also note the use of unaccustomed terms, such as the aorist passive participle of the verb ,118 which we have paraphrased as “after performing the quadrature.” The deficient mathematical information did not allow Komtino to restore a coherent text. The Hebrew translation reflects this: it is extremely faithful and, consequently, incoherent. The answer is a rough quotation of Proposition XII.7 of Euclid’s Elements (“Any prism with a triangular base is divided into three pyramids equal to one
1 in part (⅓ instead of ∕6) without changing the numerical result. It is also conceivable that B took its reading from Pet. based on the correction. If we accept the first scenario, we would have to hypothesize two stages of correction of this manuscript. It is hard to say whether the marginal correction was made by the original scribe or by a later hand—what is clearly the case for other scholia later in the text—because the codex was trimmed and the marginal notes have been severely damaged. 116 This is the value given by Silberberg, “Ein handschriftliches hebräisch-mathematisches Werk II,” p. 232, the result of his own correction. 117 As is already remarked by Heiberg, we must suppose several lacunae in the text of this passage of S. We indicate those we were able to detect by means of square brackets enclosing an ellipsis; the same applies to Komtino’s version presented below. 118 The verb is not rare, but with the prefix - it is a hapax in the entire body of ancient Greek literature.
233 Hero of Alexandria and Mordekhai Komtino another with triangular bases”), combined with the remark (phrased rather strangely) that a prism with a triangular base is half of a parallelepiped with a quadrangular base, from which one must deduce that a pyramid with a triangular base is one-sixth of the parallelepiped with a quadrangular base. The remark is correct, but does not take account of the computation performed here. To justify its application, the author of what must have originally been a marginal note later incorporated into the main text seems to suggest that one has performed the quadrature of the triangular base, which Komtino faithfully respects:
Because of the reason that119 I will mention; which is that the prism whose base is a triangle, which is a semi-square, is divided into three pyramids that have triangular bases, and so too this prism; as is demonstrated by Euclid’s proof in Book XII. And the prism is half of a square (sic: of a parallelepiped with a quadrangular base?) and is divided into three pyramids, and this is 6 ½ ⅓;120 and necessarily [the volume of] the pyramid with a triangular base has a sixth […], the base having been transformed .and we take one sixth 121(תושבת מהמרבעת) into an equal square 3. But if the triangle was isosceles, for example if the equal sides are 12 cubits, the base 8 cubits, and the inclinations of the pyramid 25 cubits, proceed as follows: Divide the base, [take] half of 8; the result is 4. [Multiply] this by itself: the result is 16. Now [multiply] one of the sides, 12, by itself: the result is 144. From this, subtract 4 [multiplied] by itself (the result is 16): 122 1 1 the remainder is 128; of which the square side is 11¼ ∕22 ∕44 cubits. This is the perpendicular on the base of the isosceles triangle. And the area: do as follows; [multiply] the perpendicular on the 1 1 1 123 base, 11¼ ∕22 ∕44, by 8: the result is 90½ ∕22. [Multiply] this by the
119 NY, fol. 64a, resumes here after the lacuna caused by the loss of a folio. 1 120 We do not know where this numeric value, not found in S, comes from; it is close to ∕6 of 41⅔ (6 × 6½ ⅓ = 41). 121 An extremely strange expression in Hebrew, a calque of . Komtino wanted to convey the nuance he detected in the prefix -, which indicates movement away from; literally: from the result of the fact of having squared the triangular base and we take one-sixth. 122 ] corr ex S; Komtino has 128. 123 ] om. S; ½ is omitted by Komtino as well.
234 Tony Lévy and Bernard Vitrac
perpendicular of the pyramid, which you find as follows: for every 1 triangle, universally, taking ½ of the perpendicular to the base, 11¼ ∕22 1 1 1 1 1 ∕44 ; the result is 5½ ∕8 ∕44 ∕88. [Multiply] this by itself: The result is 32 ∕44; and [multiply the 25] of the edge by itself: the result is 625. Subtract the 1 124 32 ∕44: the remainder is 593. From that a square side: the result is 24 ¼ ⅛ with a remainder. That is the perpendicular. 1 125 Multiply that by the area of the triangle, by 90 ½ ∕22: the result is 2207.126 From this, take the sixth […]127 of a square; the result is 367 ½ ⅓.128 This is the [volume of] the pyramid.
The first step is to calculate the height of the isosceles triangle that is the base, 1 1 1 1 namely 11¼ ∕22 ∕44, and then its area, (½)∙[8.(11¼ ∕22 ∕44)]. But the text leaves out 1 the step of dividing by two and therefore produced double the area, 90 ½ ∕22 (both S and Komtino left out the ½). The second step is to determine the height of the pyramid from the edge and the height of the base triangle, which has just been computed. Komtino reproduces the computational error of his pseudo-Heronian model (an error already noted by Schub, “A Mathematical Text,” p. 69), which is the assumption that the height of the pyramid falls on the midpoint of the height of the isosceles triangle that is the base. At the end of the problem, the same algorithm as in the previous exercises is applied, using one-sixth of the product of twice the base by the height; this has the great advantage of compensating for the error committed previously about the area of the base of the pyramid! 1 Note that this incorrect algorithm, (B, h) →B∙h → ( ∕6)∙(Bh)=V, no doubt the result of a simple error in Stereometrica II.60, is also found in the calculation of the volumes of pyramids with regular pentagonal and hexagonal bases and in that of the ridged pyramid with a triangular base (Stereometrica II.63–64 and 66, and this in both recensions of the pseudo-Heronian corpus),129 accompanied by the
124 ] S and 32¼ in Komtino. 1 125 ϙ ] ϙ S. Komtino, too, has 90 ∕22. 126 ] S. Komtino has 2199. This is not the expected product of the multiplication of 1 24 ¼ ⅛ by 90 ∕22. On the other hand, we do obtain 2199 if we multiply Komtino’s final (incorrect) result, suggesting that he worked backwards from the latter; the magnitude of the result indicated in S (82 ½) is unacceptable. 127 Lacuna indicated by Heiberg. 128 < ] < S. Komtino, too, has 366½. 129 The traditions of S and of C, which, for the stereometric problems, includes Par. gr.
235 Hero of Alexandria and Mordekhai Komtino
1 same pseudo-justification: “because it is ∕6 of a prism” ( ). It is quite probable that these are unlucky “corrections” made from the annotation of problem II.60, modifications that fortunately are not systematic (they spare not only the previous problems, II.55–57, but also II.61, 62, 65). Although he sometimes corrects computational errors in his model—when the order of magnitude is clearly impossible130—Komtino meticulously stuck to his model with regard to this mistaken algorithm in Stereometrica II.63–64 and 66 and took over the same pseudo-justification.
The Sixth Encounter: The Measurement of Round and Mixed Solids The next chapter (NY, fols. 67b7–71a4; Pet., fols. 38b–39b; B, fols. 89b–93b; P, fols. 63b–64b) deals with the measurement of spheres and round bodies (Bar Ḥiyya’s Class 3). Komtino begins with a sphere of diameter 7 and calculates its surface area and volume. Now, however strange it may seem, in the entire Heronian and pseudo-Heronian corpus there is only one example of such a problem with this numerical data: the problem edited by Heiberg as Stereometrica I.3b and found in S. Other pseudo-Heronian manuscripts have an “excerpt” (Stereometrica I.3a), but without the calculation of the surface area. Consequently, in S the problem has two diagrams, one for computing the volume (fol. 12r), and the other for computing the surface area (12v). The same is found in Komtino (NY, fol. 68a; P, fol. 64a; the diagram is not found in B). But it turns out that Bar Ḥiyya provides this same example (chapter 4, §178), although the algorithm applied there is somewhat different:
• In Stereometrica I.3b, the first step is to calculate the volume using the algorithm131 V = D → D3→ (11) ∙ (D3) → (11) ∙ (D3) / 21 And only then surface: S = D ∙ circ. 1 2 • Bar Ḥiyya first computes the surface area—S = (3+ ∕7) ∙ D —and then the volume, according to the rule V = S ∙ (D/6).
2475 and Monac. gr. 165 (both 16th c.). Codex A has only plane problems. See Héron d’Alexandrie, Metrica, pp. 439–441. 130 See above, nn. 122 (128/28) and 126 (2199/82). 131 Of course D designates the diameter of the sphere, S the surface area, V the volume, and circ the circumference of a great circle.
236 Tony Lévy and Bernard Vitrac
Komtino begins by following Bar Ḥiyya’s procedure (and employs his terminology as well). Then, having noticed that the “same” problem is found in S, he inserts what he refers to as “a different way”—a passage that is in fact merely a faithful translation of Stereometrica I.3b, followed by a series of exercises that are translation of the problems Stereometrica I.55–57 and 59–60, I.19, I.12, I.18, and I.15. This enumeration, according to Heiberg’s edition, suggests a random order; but if we return to S (fols. 12r–15r) we see that it is the sequence of problems 27–30, 32–33, and 35–38 in the collection entitled Euclid’s Geometry, which follows problem Stereometrica I.3b (= No. 27):
Stereom. I.3b = other way: see above. Stereom. I.55: calculation of the hollow space within a sphere (D = 10, thickness = 2). Stereom. I.56: measurement of a hemisphere (D = 7). Stereom. I.57: calculation of the hollow space within a hemisphere (D = 14, thickness = 2). [Stereom. I.58: measurement of the surface of the same hemisphere]. Stereom. I.59: measurement of a “shell” (quarter-sphere) (D = 14). Stereom. I.60: calculation of the hollow space within the same “shell” (D = 14, thickness = 2). Stereom. I.19: measurement of the volume of a cylinder (length = 50, d = 7) Stereom. I.12: measurement of the volume of a cone (d = 7, h = 30) Stereom. I.18: measurement of the volume of a cone (d = 42, Kl = 75); surface of the same cone.
Stereom. I.15: measurement of the volume of a truncated cone (d1 = 10,
d2 = 4, Kl = 30)
One can note that problems Stereometrica I.58 and 1.61–62 (corresponding to Euclid’s Geometry 31 and 34), left out of Komtino’s treatise, measure respectively the surfaces of the hemisphere and quarter-sphere whose volumes are calculated by problems 30 and 33. Here Komtino is interested only in the calculation of volumes, full or hollow; consequently he omits the measurement of the surface area of the hemisphere and quarter-sphere (but not of the sphere or, later, of the cone). Somewhat troubling is that interpolated among the problems corresponding
237 Hero of Alexandria and Mordekhai Komtino to Stereometrica I.55–57 and 59–60, I.19, 12, 18, 15, which are consecutive in S, are two problems (Nos. 18 and 19 in Silberberg, “Ein handschriftliches hebräisch- mathematisches Werk,” II, p. 235) relating to the volume of two complementary spherical segments in a sphere of diameter 7 and with sagittas of 2 and 5, thus respectively smaller and larger than a hemisphere and complementary in the sphere of diameter 7 measured at the start of the chapter in Stereometrica I.3b (= Euclid’s Geometry 27) and in Bar Ḥiyya §178. We found no trace of these problems either in S or in the entire Heronian and pseudo-Heronian corpus. Once again, however, a very similar problem appears in Bar Ḥiyya (§180). The next to last problem of this series has another interesting feature: whereas Stereometrica I.18 is found in both recensions of the Heronian corpus (S [I.18b] and the family of codex C (I.18a), what distinguishes the versions, other than tiny editorial differences, is the computation of the lateral surface of the cone, which is the third part of the problem in S (fol. 14v, after the height and volume), and found only in S. But the computation is erroneous, because it employs the previously calculated height of the pyramid instead of the inclination. Reference to NY (fol. 70b), Pet. (fol. 39b), and B (fol. 93a, method not pursued)—P breaks off just before this problem—shows that Komtino did translate the calculation of the lateral surface of the cone, with the same error.
Komtino and Codex S Based on our reading, we cannot avoid noting that the two essential elements of Schub’s description of Komtino’s use of his Greek source (see above, the section “Komtino’s Mathematical Treatise: The State of the Question”) seem to be incompatible. Hence we must conclude:
• Either Komtino did not use codex S (or he used other manuscripts of the Greek metrological corpus in addition to it); • Or his work on his Greek source was not merely that of a translator.
We might even adopt both explanations. However, we are convinced that the element of the description that should be retained is Komtino’s use of Seragliensis G.I.1 (S) as his copytext for the Heronian and pseudo-Heronian metrological corpus. Many arguments support this conclusion. However prudent we wish to be, their convergence ultimately leaves no doubt about this question. As Schub underscored, S is the only extant codex that contains the
238 Tony Lévy and Bernard Vitrac
Metrica. But this may not have been the case in the fifteenth century: the fall of Constantinople in 1453 accelerated the movement of Greek manuscripts to the West and the disappearance of some. There were also accidental losses in Western Europe; the fires in the Escorial Library (1671), the Protestant Seminary of Strasbourg (1871), Turin (1904), and Bologna (1944) destroyed many Greek manuscripts. However, a detailed examination of the diagrams in Metrica I.11–12 as found in S discloses a number of anomalies in the method of its copyist, who was otherwise very careful and methodical. Fortunately, they were among the diagrams that Komtino retained—and we see that the anomalies in question occur in the layout of the Hebrew text, particularly as transmitted by the New York manuscript132 and its descendants. It is implausible to postulate a second copy of the Metrica, including the same diagrammatic vagaries as S, which still existed at the end of the fifteenth century but later disappeared without a trace. What is more, Komtino did not only take over propositions I.11–16 and 17–25 from the Metrica; he also quarried several problems from the pseudo- Heronian corpus that, in our current state of knowledge, are found only in S: Geometrica 19.6, Geometrica 19.8; Stereometrica I.55–60, II.57, and II.60. We should also note, following Schub, that in Stereometrica II.60.3, Komtino applied the same mistaken algorithm to calculate the height of the pyramid. But it would be prudent to note that this evidence of a single source does not hold for some borrowings that are found in other manuscripts as well:
— Geometrica 3.1–23 is found in S, in V, in the AC family of manuscripts, and in the Geodesy. — Geometrica 17. 4bis–3-3bis is found in S and V. — Geometrica 17.1 is found in S, in Geom. 17 AC, in Geom. 21 A, and in De mensuris. — Diophanes is found in S and in Par. gr. 2448. Stereometrica I.19, 12, 18, 15 is found in S and in the family of codex C. — Stereometrica II.55 and 8 are duplicated in S (Euclid’s Geometry; Measurement of Pyramids) and are found in the family of codex C. Stereometrica II.55 is also found in De mensuris. — Stereometrica II.61–68 is found in S and in the family of codex C.
The stereometrical items do not pose a grave difficulty. Schub already emphasized
132 For details, see below, Appendix B.
239 Hero of Alexandria and Mordekhai Komtino that in his procedure for measuring a truncated cone (Stereometrica I.18) Komtino reproduced the same computational error for the lateral surface.133 As we noted above, the problem of the measurement of a truncated cone is found in other manuscripts, but not this particular computation, which is unique to S. What is more, the order of the problems in Komtino is that of S and not of the collection Stereometrica I (which must have been 12, 15, 18, 19, because this is the order in Heiberg’s edition), which also interpolates other problems. On the other hand, Schub’s invocation of a “common mathematical error” is not fully persuasive, given the relative antiquity of codex S, which may have been the archetype of several collections of problems. This is quite probably the situation for the collection entitled Stereometrica II.134 Hence the mistake in 1 several problems on measuring pyramids (confusion of ⅓ and ∕6) is found in S and in Komtino as well as in manuscripts of the codex C family. In fact, we have two reasons for concluding that Komtino took it from S:
1. Two problems are not found in the section that is common to the Greek manuscripts (Stereometrica II.55, 58, 61–68) but are in Komtino’s text (Stereometrica II.57 and 60). And S is the only codex that contains the full collection of the problems on complete or truncated pyramidal solids (Stereometrica II.55–68). 2. What is more, the manuscripts of Komtino (NY, Pet., and P) have fairly complex diagrams for these problems135 that are extremely similar to those in S, whereas those of the codex C family are rough sketches only.
We may add that the conventions of spatial representation followed in S and scrupulously respected in NY are quite arbitrary, which excludes the possibility of coincidence. Even if a copyist was not satisfied with the rough drawings in C, the probability that he would reconstruct diagrams identical to those in S is nil. Here, for example, are the diagrams for Stereometrica II.55 and 58 (Fig. 6) that have variants (Stereometrica I.30 and 32) in the family of C and its (many) descendants:
133 Schub, “A Mathematical Text,” p. 69. This is one of the arguments he used to justify his hypothesis that Komtino had been excessively faithful to S. 134 See Héron d’Alexandrie, Metrica, pp. 471–474. 135 On the other hand, B does not have diagrams for Stereometrica II.60–62 and 66, and those for II. 55, 57–58, 65, 68 are very sketchy and drawn freehand.
240 Tony Lévy and Bernard Vitrac
S, Stereometrica II.55, fol. 55r JTSL 2639, fol. 63b136 C, Stereometrica I.30a, fol. apud Bruins [= Exercise 1, 100r (variant of Stereometrica (Brill, 1964, supra note 31), Silberberg, pp. 231–232] II.55) p. 105 Courtesy of The Library of the Jewish Theological Seminary
S, Stereometrica II.58, fol. 56r Pet., fol. 36a; C, Stereometrica I.30a, fol. apud Bruins lost in NY 100v (variant of Stereometrica (Brill, 1964, supra note 31), II.58) p.107
Figure 6
136 In Pet. (fol. 36a), the window provided for the diagram of this problem has been left empty.
241
Hero of Alexandria and Mordekhai Komtino
Similarly, S is the sole witness of the full sequence Geometrica 17.4bis +17.1 +17.3 +17.3bis. What is more, in other places with common items, we have observed mathematical or philological variants, but Komtino’s translations are always in accordance with S. The only instance that does not permit a decision merely by a comparison of the texts relates to the short treatise by Diophanes, which exists both in S and in Par. gr. 2448. Unfortunately, the section that interests us (Diophanes 10–17) does not seem to have any variant that can distinguish between the two codices and that we might hope to find in Komtino. But why would he have used a different source for this one passage? What is more, even if there are very few textual variants between our two Greek exemplars that might survive translation, we nevertheless observe that they treat diagrams very differently. In Par. gr. 2448, only the pentagon and hexagon are diagrammed; in S and Komtino (NY), each of the eight measurements has its own carefully drawn diagram. Hence there is little doubt that Komtino took the passage of Diophanes from S. The exclusive dependence on this manuscript and no other representative of the pseudo-Heronian corpus is corroborated by the fact that when Komtino’s text includes problems found in that corpus but not in S, they are also found in the treatise by Abraham Bar Ḥiyya, Komtino’s other source. Hence Komtino did not have to have recourse to some other Greek manuscript to find them. If we accept this conclusion, and recall what we found about the translation of the line numbering in Geometrica 3.4, of Metrica I.11, 18, and 20–25 (omission of the numerical procedure), and of Metrica I.12, 13, 14b, and 16 (switching the order of the proof and the numerical procedure), it becomes quite clear that Komtino’s is not a literal translation, but a version that was reworked, especially in the proofs. This can be seen from the following as well:
• In Komtino’s treatment of excerpts from the collection known as Euclid’s Geometry, where he separated the plane and stereometric sections to suit his own scheme; • In the omission of an alternative algorithm for the measurement of the regular hexagon in Diophanes 11 (a simple variation in the expression of fractions), but retained for the measurement of the decagon in Diophanes 15 (because the second algorithm is considered to be more accurate than the first);
242 Tony Lévy and Bernard Vitrac
• In the omission of the question about the diameter of the circumscribed circle in Diophanes 10, 13, 15; • In the deletion of problems Stereometrica II.56 (a near duplicate of II.55) and II.59 (an algorithm that is incorrect or only approximate).
Komtino takes a more conservative attitude towards his model in the sequence of exercises in §3. Either he had grown weary, or he was less interested because these were only exercises.
Conclusions Nothing of what Komtino did to his sources required great skill in mathematics, but it does suppose accurate comprehension of the source texts and a literary ambition that goes beyond the production of a simple compilation. But that is more or less what Silberberg’s and Schub’s evaluation reduced Komtino’s labors to, and their judgment seems to be a bit severe. The former (1906, p. 237) insisted on the modest nature of the project, “envisioned first of all for daily use” (vor allem für die tägliche Praxis berechnet). The latter (1932, pp. 54–55) placed our author in a Byzantine-Greek tradition whose mathematical endeavors, to judge by the geometries of Pediasimos or Isaac Argyrus, were limited to a few glosses on arithmetic and geometry. In defense of our forerunners, we observe that the vast majority of historians of mathematics in the late nineteenth and early twentieth centuries tended to make mathematical originality the sole touchstone for their judgments and to attach too high a value to the demonstrative stream of the Greek tradition and its heirs. Hence an author who failed to make a new contribution or adopted an algorithmic approach, as is common in metrological geometry, was not rated very highly. In that period, that is precisely what happened to Hero of Alexandria himself.137 For its author’s era, and the field of metrological geometry (“practical geometry”), Komtino’s treatise, with its massive borrowings from Hero’s Metrica, the various pseudo-Heronian collections ( [Euclid’s Geometry], Diophanes, Measurement of Pyramids), collated with Abraham Bar Ḥiyya’s work, the product of a different tradition, but which also demonstrates a constant regard for proofs, strikes us as a quite exceptional essay.
137 Héron d’Alexandrie, Metrica, pp. 26–31.
243 Hero of Alexandria and Mordekhai Komtino
Thanks to his choice of sources, Komtino was able to combine two qualities:
• An insistence on demonstrations: this is frequently lacking in collections of geodesic problems, which consequently must offer multiple examples so that readers can comprehend the generality of the procedures they present. • An effective procedure in the measurement of different types of figures, such as polygons and pyramids, where other authors, more abstract in their approach (and this may apply, depending on the topic, to Hero and to Abraham Bar Ḥiyya) were often content with rather general indications.
Finding the proper balance between these two objectives is not necessarily easy—it also depends on the intended readership—but, for Komtino, the attempt to do so seems to have led him to switch among his sources. Had he merely wished to produce a compilation or epitome, he needed only to duplicate parts of the work by Abraham Bar Ḥiyya.
244 Tony Lévy and Bernard Vitrac Appendix A: The Structure of Book II (the geometry) of Komtino’s work138
I. The measurement of plane figures I.1 Terminology [NY, fols. 30b1–32b9; Pet., fols. 19a–20b; B, fols. 36a–38b; P, fols. 42a–43a] I.1.1 Introduction. Free paraphrase of the definitions in Book I of Euclid’s Elements,139 cited explicitly by title and author; inversion of the rectilinear and non-rectilinear types. I.1.2 Transitional section on the notion of the “area of a figure”. I.1.3 New mention of Euclid (treatise on geodesy): translation of a lexicon of plane geometry (≈ Pseudo-Hero, Geometrica 3.1–23). I.2 Measurement of squares, rectangles, and their diagonals [NY, fols. 32b9–36a21; Pet., fols. 20b12–22b6; B, fols. 38b–43b; P, fols. 43a–45b] I.2.1 Introduction. Quotation from the “second part of the Arithmetic composed by Nicomachus of Gerasa, explaining that the triangle is the basic figure (the “root”) for the measurement of all the others” (cf. Nic. Ar. II, vii, 4).140 I.2.2 Area of a square with a side of 10 cubits I.2.3 Area of the rectangle (10, 5). Computational rules justified by means of quadrillage. I.2.4 Area of a rhombus with sides 10 and diagonals of 16 and 12. I.2.5 Measurement of the diagonal of a square I.2.6 Measurement of the diagonal of a rectangle. Quotation from Eucl. El. I.47. I.3 Measurement of triangles by their types [NY, fols. 36a22–40b3; Pet., fols. 22b6–24b; B, fols. 43b–49b; P, fols. 45b–48a] I.3.1 Division of triangles into equilateral, isosceles, and scalene. Simple
138 We have added a decimal numbering of the subsections, not found in the original text, to facilitate cross-references. 139 Here Komtino inserts the definition of the segments of a circle, originally found in Book III, but frequently moved closer to those of the circle and semicircle, including in most Greek manuscripts of the Elements. 140 Cf. Nicomachus, Introduction to Arithmetic II, vii, 4, 87.7–19 Hoche:
(“Thus the triangle is the most fundamental and most elementary of the plane figures. ... It is thus a component element of all the others, but none of them are components of it”). Note that Nicomachus stresses the composition of figures, not their measurement.
245 Hero of Alexandria and Mordekhai Komtino
justification of the algorithms that are valid for all types of triangle: A = h ∙ (B/2) or A = B ∙ (h/2). Demonstration of these two algorithms for an equilateral triangle. Section strongly inspired by Abraham Bar Ḥiyya, §§58–60. I.3.2 Method of “apotomes” for a scalene triangle141 with sides 16, 18, 20. Calculation of two apotomes (5½ ⅛, 10 ⅜), but not of the height, which cannot be expressed, nor, a fortiori, of the area.142 I.3.3 Method of “apotomes” for the “canonic” scalene triangle with sides 13, 14, 15. Cf. Bar Ḥiyya §64, but with major variations between the two Hebrew authors.143
141 A scalene triangle requires to a more complex procedure, because the height does not fall on the midpoint of the base, but divides it into unequal segments (if indeed it falls inside the triangle). To compute the height of the triangle (by the hypotenuse theorem, Elements 1.47) one must know the length of the segments the height marks off on the base. In the pseudo-Heronian corpus, these segments are called the (large and small) apotomes. 142 Edited by Silberberg, “Ein handschriftliches hebräisch-mathematisches Werk II,” pp. 219–222, with a German translation. Although one may wonder whether there has been some corruption of the text here, we have verified that the same somewhat incongruous numerical data are found in the manuscripts and that Guttmann’s critical apparatus does not indicate any variant for the data of this triangle, which, in any single manuscript, might have misled a reader. Note that after the example of the canonic acute-angle triangle (13, 14, 15), Bar Ḥiyya (§67) inserts a right-angle triangle (6, 8, 10), which is 10 less on each side than (16, 18, 20). Could there have been some confusion here between the symbol for 10 and, perhaps, a unit of measurement? of (מעמד) Abraham Bar Ḥiyya introduced a detailed terminology for “the position 143 of the latter (that is, the foot of the height), and the (גבול) ”the height,” the “boundary between the boundary of the position of the height from the (long and (מרחק) interval“ short) sides,” which correspond to the apotomes of the pseudo-Heronian corpus. An almost identical same terminology is found in Komtino, who thus was inspired by his predecessor, but did not reproduce his text exactly, as indicated in two ways: Bar Ḥiyya and Komtino used the same canonical triangle ABG with sides (13, 14, 15), but reversing the long and short sides (AB = 13, AG = 15 for Bar Ḥiyya; AB = 15, AG = 13 for Komtino). To keep the lettering found in the text, Komtino first deals with determining the “boundary of the position of the height from the short side (AG)” and then from the long side (AB), whereas Bar Ḥiyya proceeds in the opposite way. There is no trace in Bar Ḥiyya (nor in S or elsewhere in the Heronian corpus) of the uncompleted problem of §I.3.2.
246 Tony Lévy and Bernard Vitrac
I.3.4 Method of “apotomes” for the obtuse-angle scalene triangle with sides 4, 13, 15. Cf. Bar Ḥiyya, §§69–70.144 I.4. Measurement of “slanted figures” by their types [NY, fols. 40b3–44a2; Pet., fols. 25a–27a; B, fols. 49b–54b; P, fols. 48a–50b] I.4.1 Introduction. Types of trapezia. I.4.2 Area of the isosceles trapezium (18, 8, 13, 13) with a proof. Cf. Abraham Bar Ḥiyya, §§76–78. I.4.3 Height (called the “perpendicular”) and area of the isosceles trapezium (AB = GD = 13, AD = 6, BG = 16). Cf. Metrica I.11. I.4.4 Height (called the “perpendicular”) and area of the acute-angle trapezium (AB = 13, GD = 20, AD = 6, BG = 27). Cf. Metrica I.12. I.4.5 Height (called the “perpendicular”) and area of the obtuse-angle trapezium (AB = 13, GD = 20, AG = 6, BD = 17). Cf. Metrica I.13. I.4.6 Area of the quadrilateral ABGD with a right angle G but no parallel sides, AB = 13, BG = 10, GD = 20, DA = 17. Cf. Metrica I.14b. I.4.7 Proof that the “perpendicular” from to is given. Cf. Metrica I.15. I.4.8 Measurement of the area of another quadrilateral with a right angle G but no parallel sides, AB = 13, BG = 10, GD = 8, DA = 25. Cf. Metrica I.16. I.5. Measurement of circular figures [NY, fols. 44a2–48a2; Pet., fols. 27a–29a; B, fols. 54b–60b; P, fols. 50b–52b] I.5.1 Introduction (review of the definition of a circle; Types of circular figures; presentation of the structure of the section). I.5.2 Measurement of the area of the circle; five algorithms. Cf. Abraham Bar Ḥiyya, §95; Pseudo-Hero, Geom. 17.3a +17.3bis, a +17.4bis, a +17.1a (S, fols. 9v15–10r19). Two mentions of Ptolemy’s Almagest. 1.5.3 Measurement of a semicircle. 1.5.4 Determination of the diameter of a circle containing a segment with base (b) and sagitta (s) given (b = 8, s = 2 thus d = 10). Quotation from Eucl. El. III.35. Cf. Bar Ḥiyya, §106. 1.5.5 Measurement of the area of the segment of a circle less than a circle with base 12 and sagitta 4;
144 Both of them ground the procedure of apotomes on Euclid’s Elements II.12 for an obtuse-angle triangle (Bar Ḥiyya, §69; Silberberg, “Ein handschriftliches hebräisch- mathematisches Werk II,” p. 222); for the acute-angle triangle, however, Komtino draws directly on Elements II.13, whereas Bar Ḥiyya (§64) derives the result from Euclid’s Elements I.47 and II.4.
247 Hero of Alexandria and Mordekhai Komtino
(b, s) → [(½)(b + s)] ∙ s + (1/14)(b/2)2 = A. Cf. Geom. 19.8a, S, fol. 12r.145 1.5.6 Measurement of the arc (circ.) of the segment of a circle less than a semicircle with base 40 and sagitta 10; (b, s) → [(b + s) – (1/4)(b + s)] + (1/4) ∙ [(b + s) – (1/4)(b + s)] = circ. Cf. Geom. 19.6a, S, fol. 11r. 1.5.7 Measurement of a circular sector. Cf. Bar Ḥiyya, §109. 1.5.8 Measurement of two lunes. Cf. Bar Ḥiyya, §110. 1.5.9 Measurement of a lens. Cf. Bar Ḥiyya, §111. Note that Komtino does not include the measurement of the area of an ellipse (Bar Ḥiyya, §112) or the specific table of chords presented and employed by Bar Ḥiyya in §§113–123. I.6. Measurement of regular polygons [NY, fols. 48a2–54a17; Pet., fols. 29a–31a; B, fols. 60b–69a1; P, fols. 52b–56a] I.6.1 Introduction. General remark on the possible measurement of polygons by triangulation. Cf. Bar Ḥiyya, §126. I.6.2 Algorithms for measuring the area of eight regular polygons, from the pentagon to the dodecagon. Cf. Diophanes, 10–17. Same alternative algorithm for the decagon, here said to be more accurate. I.6.3 Transitional link, procedures and proofs (text quoted above, Fourth Encounter). I.6.4 Proofs of measurement of eight regular polygons, preceded by two lemmata for triangles. Cf. Metrica I.17–25. II. Divisions of plane figures II.1 Divisions of triangles according to their types (equilateral or isosceles triangles, divided into 2 or n equal parts by dividing the base; triangle into 2 equal parts by a line parallel to the base, into 3 by a line parallel to the base + dichotomy; a scalene triangle into 2 or 3 equal parts). Cf. Bar Ḥiyya, §§129–134 [NY, fols. 54a18–56a4; Pet., fols. 32a–33a; B, fols. 69a–71b; P, fols. 56a–57a] II.2 Divisions of quadrangles; division of a square into 2, 3, 4 equal parts; division of a rectangle; division of a rhombus into 4 equal parts; division of a parallelogram into 4 equal parts; division of a trapezium into 3 equal parts. Cf. Bar Ḥiyya, §§135–148 [NY, fols. 56a5–60a6; Pet., fols. 33a–35a; B, fols. 71b–77a; P, fols. 57a–59a] II.3 Division of the circle into n (in fact, 3) equal parts by dividing the circumference.
145 This and the next measurement are not found in Bar Ḥiyya’s exposition (§§107–108). He calculates the area of the segments of a circle from the length of their arcs, without explaining how to obtain the latter from rectilinear data (base and height).
248 Tony Lévy and Bernard Vitrac
Cf. Bar Ḥiyya, §151 [NY, fol. 60a6–23; Pet., fol. 35a; B, fol. 77a–b; P, fol. 59a] None of these division problems are found in Metrica III. III. The measurement of solid figures III.1 Review of terminology [NY, fol. 60a24–60b18; Pet., fol. 35a(-1)–b; B, fols. 77b11–78a7; P, fol. 59a20–59b4] III.2 Algorithms for measuring volumes [NY, fols. 61b19–63b7; Pet., fols. 35b8–36a? (damaged); B, fols. 78a7–82a17; P, fols. 59b4–61a10] III.2.1 Volume of a cube with sides of 10 cubits and of a parallelepiped (10, 15, 20). Observation about the solid cubit. Calculation of their diagonals. Parallelepipeds with a rhomboid, triangular, pentagonal, hexagonal, or circular (= cylinder) base. Cf. Bar Ḥiyya, §§156, 156a in part, 159. III.2.2 Volumes of stepped solid. General procedure. Method for finding the height of figures. Cf. Bar Ḥiyya, §§164–165, 167–167a–168. III.2.3 Volumes of solids “truncated at the top.” Measurement by difference. Another procedure. Examples; Volume of the truncated pyramid with a square base
(C4 = 6, c4 = 4, h = 10); volume of the truncated cone with (d1 = 4, d2 = 2, h = 12); remark about the measurement of barrel-shaped solids (which can be assimilated to the sum of two conical frustra); volume of the truncated
pyramid with equilateral triangle base (C3 = 8, c3 = 4, h = 6); general procedure for measuring truncated solids. Cf. Bar Ḥiyya, §§172, 174, 175, 176, 177. III.3 Transition. Exercises. Examples of the measurement of pyramids and truncated solids. [NY, fols. 63b9–67b6; Pet., fols. 36a–38b2; B, fols. 82a18–89b1; P, fols. 61a–63b]
III.3.1 Volume of the pyramid with a square base (C4 = 24, Kl = 18). Cf. Stereometrica II.55.
III.3.2 Volume of the pyramid with a square base (C4 = 12, Kl = 36). Cf. Stereometrica II.57.
III.3.3 Volume of the truncated pyramid with a square base (C4 = 10, c4 = 2, Kl = 9). Cf. Stereometrica II.58. III.3.4 Volume of the pyramid with a right-triangle base (h = 25, sides of the right angle: 4, 5). Cf. Stereometrica II.60.1–2. III.3.5 Volume of the pyramid with an isosceles triangle base [Base = triangle (12, 12, 8), Kl = 25]. Cf. Stereometrica II.60.3–5. III.3.6 Volume of the pyramid with an obtuse-angle or acute-angle triangular base. Treatment by analogy. Cf. Stereometrica II.60.6. III.3.7 Volume of the pyramid with a right-triangle base [Base = right angle (6, 8, 10), Kl = 13]. Cf. Stereometrica II.61.
249 Hero of Alexandria and Mordekhai Komtino
III.3.8 Volume of the pyramid with an equilateral triangle base (C3 = 30, Kl = 20). Cf. Stereometrica II.62.
III.3.9 Volume of the pyramid with a pentagonal base (C5 = 12, Kl = 35). Cf. Stereometrica II.63.
III.3.10 Volume of the pyramid with a hexagonal base he (C6 = 12, Kl = 35). Cf. Stereometrica II.64.
III.3.11 Volume of the pyramid with an octagonal base (C8 = 10, Kl = 15). Cf. Stereometrica II.65.
III.3.11 Volume of the fluted pyramid with an equilateral-triangle base (C3 = 10, depth of the grooves = 2, Kl = 20). Cf. Stereometrica II.66. III.3.12 Quotation from Euclid’s El. XII.7. General rule for the calculation of the volume of a pyramid. Cf. Stereometrica II.67. III.3.13 Volume of the bomiskos [h = 50, base = rectangle (24, 16), opposite face = rectangle (12, 8)]. Cf. Stereometrica II.68. III.4 Measurement of round bodies [NY, fols. 67b7–71a4; Pet., fols. 38b3–39b19; B, fols. 89b2–93b11; P, fols. 63b14–64b; codex P breaks off here] III.4.1 Surface and volume of the sphere of diameter 7. Cf. Abraham Bar Ḥiyya, §178. III.4.2 Exercises. Examples of the measurement of round bodies. III.4.3 Calculation of the hollow space within a sphere (D = 10, thickness = 2). Cf. Stereometrica I.55. III.4.4 Measurement of the volume of a hemisphere (D = 7). Cf. Stereometrica I.56. III.4.5 Calculation of the hollow space within a hemisphere (D = 14, thickness = 2). Cf. Stereometrica I.57. III.4.6 Measurement of the volume of the “shell” (quarter-sphere) (D = 14). Cf. Stereometrica I.59. III.4.7 Calculation of the hollow space within the same shell (D = 14, thickness = 2). Cf. Stereometrica I.60. III.4.8 Measurement of the volume of a spherical segment smaller than a hemisphere. Cf. Bar Ḥiyya, §180. III.4.9 Measurement of the volume of a spherical segment larger than a hemisphere. Cf. Bar Ḥiyya, §180. III.4.10 Measurement of the volume of a cylinder (length = 50, d = 7). Stereometrica I.19. III.4.11 Measurement of the volume of a cone (d = 7, h = 30). Stereometrica I.12. III.4.12 Measurement of the volume and the surface of a cone (d = 42, Kl = 75). Stereometrica I.18.
250 Tony Lévy and Bernard Vitrac
III.4.13 Measurement of the volume of a truncated cone (d1 = 10, d2 = 4, Kl= 30). Stereometrica I.15. IV. Divisions of solid figures [NY, fols. 71a5–72b7; Pet., fols. 39b19–40a (end of the codex); B, fols. 93b1–95a (end of the codex); P, deest]. Komtino remarks that the divisions of solid figures are of minor important in practice, and chiefly for commerce. IV.1 Division of the cube into 2, 3, or 4 equal parts IV.2 Division of a prism with a quadrangular or triangular base into 2 equal parts IV.3 Division of a sphere and hemisphere into n parts by division of a great circle (cf. II.II.3) None of these Division problems are found in Metrica III.
251 Hero of Alexandria and Mordekhai Komtino Appendix B: Diagrammatic Anomalies in S, Metrica I.11–12 This appendix looks at the situation of the diagrams in the Hebrew translation of Metrica I.11–12. Here there is an anomaly that can be explained in part by the atypical treatment of the diagrams for these two propositions in Greek codex S. A comparison with manuscript NY reflects the relations among the several copies of Komtino’s text. We begin with S. In S, each proposition of the Metrica has a carefully drawn diagram inserted in a window left for it at the end of the proposition, sometimes opposite the first lines of the next proposition. For Metrica I.11 (isosceles trapezium) and 1.12 (acute-angle trapezium), we find such diagrams in the space between I.11 and 12 and I.12 and 13, respectively:
Metr. I.11, S, fol. 73v Metr. I.12, S, fol. 74r
K A M˚ M K A ] [ ] [ ]M[
IB IA
]K[ ] [ H I H
]K [
I
B Λ Z I E N ] [ B Λ Z E N
The lettering is exactly the same for the two diagrams. In both trapezia, = 13, = 6 (the height is thus the same, 12). But the bases and the sides are different, respectively 16, 13 and 27, 20. As a result, these drawings are both very similar and clearly wrong146
• In the diagram for I.11, we have = = 5 < = 6 (a problem of scale), = 8½. • Points H and , the respective midpoints of and , are not placed as such.
146 The lettered diagrams of the Greek mathematical texts preserved are not accurate, especially with regard to respecting proportions. From time to time, however, a very large deviation from “reality” is perceived and leads, as here, to a corrected diagram.
252 Tony Lévy and Bernard Vitrac
• In the diagram for I.12, we have = 5, = 6, = 16, and = 14 (!). As a consequence, point , which is part of segment in I.11, must be part of segment in I.12, but is not drawn as such. • Similarly, in the diagram on the left (I.11), the (= 20) written on by a later hand leads to or was suggested by an error: that this is the diagram for I.12, because here = = = 13. • In the diagram on the left, the indication (= 11) along segment M may suggest that this is its length; in fact, M = 6. It is simply a reminder that this a diagram for Metrica I.11. • The insertion of this brief indication (even though the diagram is at the standard location according to the procedure followed by the copyist Ephrem) may have been made necessary by the fact that the relative “error” of the diagram proposed for I.11 led to the drawing of an additional diagram, inserted in a window in the middle of the proposition (the middle of fol. 73r), which is quite atypical in the format of the manuscript.
The numerical values indicated on the diagram are correct; points H and are indeed the respective midpoints of and (see below).
We would add two remarks: Extra diagram for Metr. I.11
• It is astonishing that the more serious error K A ] [ M in the diagram for I.12 was never corrected.
] I [ H ] I [
]I ]IB[ [
B Λ Z ]E[ E N ]E[
Between the two diagrams provided for Proposition I.11 there intervenes text that states the result of the problem (“and it is equal to trapezium : trapezium is thus
253 Hero of Alexandria and Mordekhai Komtino
also 132 units”), followed by the numerical synthesis that Komtino omitted.147
• Now if we turn to the Hebrew manuscript NY, we find on fol. 41b, almost in the middle of the version of the text corresponding to Metrica I.11, a diagram corresponding to that of problem Metrica I.12 and, juxtaposed with it, in the far margin, a second drawing corresponding to the one just mentioned, the corrected diagram inserted in the middle of Metrica I.11 in Greek (Fig. 7).
Diagram for Metr. I.12 Diagram for Metr. I.11
Figure 7: MS JTSL 2639, fol. 41b Courtesy of The Library of the Jewish Theological Seminary
147 We would add that the text itself of Metrica I.11 does not actually follow the orthodox Heronian scheme (geometric analysis and numerical synthesis) explained above (the Second Encounter). The calculations are performed as the analysis proceeds, whereas they should be performed only in the numerical synthesis (Héron d’Alexandrie, Metrica, pp. 174.18–22). As a result, the answer to the problem is already known at the end of the analysis and the numerical synthesis is of no use. This form of anticipation—no doubt due to the modification of the text in an early stage of its transmission—and the anomaly in the treatment of diagrams discussed here, explain why Komtino did not reproduce the numerical synthesis and preferred to modify the end of his translation instead.
254 Tony Lévy and Bernard Vitrac
That the main diagram (despite its location) is that for Metrica I.12 is clearly shown by the numerical values 27 (larger base) and 20 (side GD) and, especially, by the fact that point E lies, correctly, in segment NG and not in , as in the three Greek diagrams reproduced above. On the diagram in the margin, now almost illegible, one can make out the value (13) of the oblique lines AB, AH, and DG; point H (the letter is almost wiped out, but the others are still visible and the identification is thus fairly certain by means of elimination) is part of segment , and points Ḥ and Ṭ are the respective midpoints of AB and GD. So this is the diagram corrected for Metrica I.11 in the Greek (the first in the order of the text). Komtino’s work, to the extent that we have been able to reconstitute it on the basis of the New York manuscript, is ambivalent.
• Having decided not to retain the numerical synthesis of Metrica I.11, he found in his model two “consecutive” diagrams (fols. 73r and 73v); what is more, in S, the second of these diagrams resembles (a bit too much) that of Metrica I.12, for which the relevant diagram is found at the end of the problem (fol. 74r). The last named has the merit of being correct for the numerical values that accompany it, except for the position of point E and the proportions between the sides, whereas the second of the diagrams for Metrica I.11, if associated with I.12, is wrong in both respects. • So Komtino eliminated it. He copied the diagram for I.12 and corrected the location of point E as is clear from the annotation inserted at the .(”this has been corrected“) זאת מוגהת :center of the trapezium • The ambivalence of his work had to do with the location of the diagram: in the middle of the text of the version corresponding to I.11, in other words within several lines of where the corrected diagram was inserted in the Greek text. For the marginal diagram, this did not cause any confusion; but it made the main diagram appear six lines of text ahead of the start of the corresponding version in I.12. • To be kind, we should also observe that what was two separate sections in codex S (marginal alphabetic numbering of the sections and diagrams; bold initial letters; the separator function played by the diagrams placed at the end of each propositions, except for I.11) is rendered by Komtino as two successive figures of the same measurement problem (that of “slanted figures”).148
148 This also holds true for his version of Metrica I.13 (obtuse-angle trapezium). But the linkage of text and diagram for the Hebrew equivalent of Metrica 1.13–16 poses no problem.
255 Hero of Alexandria and Mordekhai Komtino
All that remained was for a copyist or reader to be troubled by this pair of diagrams and consequently to switch the diagrams of Metrica I.11 and 12. This is what happened in Pet. (fol. 25b; see Fig. 8) and P (fols. 48b–49a).
• Both of them insert, in the windows in the text corresponding to I.11, Komtino’s corrected diagram for I.12 (Pet., fol. 25b, window at top right; P, fol. 48b, window at lower left, with a minor mistake in the lettering for point ). • Then, where the texts corresponding to Metrica I.12 and 13 meet (Pet., fol. 25b, window at the lower right; P, fol. 49a, window in the upper third on the left), we have a diagram corresponding to Metrica I.11, with significant differences. • Finally, everything returns to “normal” in the diagram for Metrica I.13 (Pet., fol. 25b, lower margin; P, fol. 49a, window in the second third of the text, on the left)
MS Pet., fol. 25b 1st window 2nd window ➛
Figure 8
Note that the diagram in the second window, which is supposed to reproduce that in the margin of NY, has several noteworthy features: It switches the lettering of the vertical lines of the rectangles (KHL and MQN [only the letters KE and MQ are legible]). It makes the opposite error of S for I.12: here H should lie on ZN and not NG. The corresponding diagram in the Paris manuscript is even more sketchy and incomplete.
256 Tony Lévy and Bernard Vitrac
MS Pet., fol. 25b Lower margin
In B, perhaps because the copyist perceived that there were problems, the blank space left for the diagram corresponding to Metrica I.12 (fol. 51a) was never filled in; that corresponding to Metrica I.11 is absent, and that corresponding to Metrica I.13 is drawn at the bottom of fol. 51b. Once again we see that the manuscript NY is not only the oldest exemplar, but also the most faithful to its model. As we explain in Appendix C, it is probably the archetype of the tradition of Komtino’s text, even if the diagrammatic anomaly just discussed was certainly to be found in the author’s autograph copy, reflecting S. In response to a reasonable question posed by an anonymous referee concerning these diagrams, we note as follows: the close correspondence between the diagrams in S and in NY exists for all the sections Komtino took from the Greek codex and not only for the examples presented here (in this appendix as well as in our discussion of the problems of regular polygons and pyramids). This strict correspondence does not exist in later copies, dating from the sixteenth century onward, notably the Berlin and London manuscripts, where the windows left for diagrams often remain empty.
257 Hero of Alexandria and Mordekhai Komtino Appendix C: On the Manuscript Tradition of Komtino’s geometry For the present study we made extensive use of four of the eight manuscripts of Komtino’s work containing the geometrical section:
NY: New York, Jewish Theological Seminary 2639 (1478) B: Berlin, Or. Qu. 308 (15th–16th c.) Pet.: St. Petersburg, Firkowitz Evr I 320b, (1495) P: Paris, BNF hébr. 1031/3 (15th-16th c.) When necessary, we consulted three other manuscripts: L: London, Brit. Lib. Add. 27107 Alm 213 (16th c.) Pet. 343/4: St. Petersburg, Firkowitz Evr I 343 + 344 (1484/85) NY 2696: New York, JTS 2696 (Adler 2785) (19th c.).
A comparison of the versions of the sections of the text of Heronian and pseudo-Heronian origin that constitute a significant proportion of sections 1 and 3 of Komtino’s geometry reveal relationships among the manuscripts that more complete and detailed comparisons could flesh out on the road to a critical edition. We believe we can state as follows:
1. All the manuscripts derive from a single archetype that is not Komtino’s autograph. 2. It is quite probable that this archetype is NY. 3. The text was revised and all the other manuscripts depend on this reworking (which must therefore date between 1478 and 1485).
The first point follows from the fact that all the manuscripts present a corrupt text in the section on the measurement of the circle (see above, n.70) and in the Hebrew version of Metrica I.17 (see above, n. 92), which can hardly be attributed to Komtino’s autograph. What is more, it is obvious that the codex NY is not this autograph, because the colophon (cited above, at n. 29) gives the name of the copyist and even notes that it is a second copy. Establishing that NY is in fact the archetype would require much more systematic collation, but we have noted in passing several points that support this; for example, the possible descent of the corrupt text in the translation of
258 Tony Lévy and Bernard Vitrac
Geometrica 3.3 in manuscripts Pet. 343/4, Pet., B, P, and L from NY (see above, n. 51) or the problems with the diagrams in the translation of Metrica I.11–12 (Appendix B). To this we can add a third item: the sequence of the exercises for the measurement of the different types of pyramids and truncated solids, taken from Stereometrica II (§III.3; see above, Fifth Encounter) ends with a statement of the name of the solid that is found in Stereometrica II.68: it is a bomiskos (small altar), a term that is rather rare in this corpus.149 In some manuscripts this ;מישקו term—Komtino states explicitly that it is a Greek word—is truncated to .in full בומישקו only NY (fol. 67a) has We have underscored that NY is also the only manuscript that presents the different species of triangles in the same order as the Greek codex S, whereas all the other manuscripts use the Euclidean order (see above, at n. 54). This reinforces the thesis that NY is the archetype of the tradition and even more so that the text has been revised. Other points suggest that the editorial alterations continued as follows: First there emerged the correction of the difficult expression “the power of its power of the square of G,” used by Komtino (NY) to render three of the four occurrences of (three of which are distorted in S). It simplifies them into “its power of the square of BG,”150 which can be interpreted either as a mathematical correction (eliminating eight-dimensional entities) or making Komtino’s text uniform, on the basis of the first occurrence. This correction occurs in Pet. 343/4, Pet., P, and L, as well as New York 2696.151 Here are the variants:
149 Except for this problem (where the term appears three times), there is only one other use of the term, in pseudo-Hero’s Definition 114 (70.6–7 Heiberg), which states that bomiskos is another name for the spheniskos. 150 The lack of symmetry in Hebrew between the two occurrences of “power” indicated by allows us to assert unequivocally that in the entire family of כח כחו של.. the Hebraism manuscripts it was the first occurrence that was dropped and not second. 151 But not in the Berlin manuscript (fol. 63b). In this codex, two lacunae, no doubt caused by the scribe’s eye jumping between the same term, swallowed up two of the four occurrences of “the power of its power of the square of BG.” However it remains faithful to the readings of New York. Similarly, a lacuna swallowed up the fourth occurrence in the Paris manuscript.
259 Hero of Alexandria and Mordekhai Komtino
Thus the power of square of BG to square אם כן כח מרבע בג152 אצל מרבע בג על מרבע (v) ,of [B]G [multiplied] by square of AD אד,153 יחסו כיחס ד אל ג, שהוא כיחס יו אל יב. its ratio is the same as the ratio of 4 to 3, which is the same as the ratio of 16 to 12;
And the square of BG by the square of ומרבע בג על אד הוא ככפל השטח הנצב הזויות (vi) AD, is the same as the product of the אשר יקיפו בו אד בג על עצמו, כלומר שני משלשים rectangular surface enclosed by AD, BG על עצמם [multiplied] by itself, that is, two triangles by themselves;
If so, the power of its power of square אם כן כח כחו של מרבע בג154 על כפל שני (vii) of BG to the product of two triangles by משולשים על עצמם, יחסם כיחס יו אל יב, themselves, their ratio is the same as the ratio of 16 to 12;
And the product of the two triangles by וכפל שני משולשים על עצמם הוא ארבעה כפלי (viii) themselves is 4 times the product of a משלש אחד על עצמו, triangle by itself;
If so, the power of its power of square of אם כן כח כחו של מרבע בג155 אצל כפל משלש (ix) ,BG to the product of a triangle by itself אחד על עצמו156, יחסו כיחס יו אל ג. the ratio is the same as the ratio of 16 to 3.
And the power of its power of square of וכח כחו של מרבע בג157 ידוע, (x) BG is known.
Fragment Pet. 343/4 presents a somewhat dicier situation to interpret: at the first occurrence of “the power of its power of the square of BG,” it has the same simplified text as Pet., P, and L; in the next two instances, though, the text is that of NY, but with indications of a correction that seem to reflect acceptance of the simpler version. We can imagine the following scenario: the copyist “automatically” corrected the second occurrence to make it coincide with the
.NY; B (63b) omitted [כח מרבע בג אצל] 152 אם כן כח התשבורת של] ;NY, Pet., Pet.343/4, L [אם כן כח מרבע בג אצל מרבע בג על מרבע אד] 153 .(P (54a [מרובע בג אצל מרבע בג על מרבע אד .omitted Pet.343/4, Pet., P, L, NY 2696 [כח] ;NY, B [אם כן כח כחו של מרבע בג] 154 is overbarred in the ”כח“ NY, Pet. 343/4 (the first occurrence of [אם כן כח כחו של מרבע בג] 155 .Pet., P, L, NY 2696 [אם כן כחו של מרבע בג] ;(MS—the normal way to indicate a deletion .NY; B omitted [אם כן כח כחו של מרבע בג אצל כפל משלש אחד על עצמו] 156 ”כח“ NY, Pet. 343/4 (as previously, the first occurrence of [וכח כחו של מרבע בג ידוע] 157 is overbarred in the MS, indicating a deletion; besides, the copyist inserted before the וכחו של מרבע בג] ;B ,(.”וכחו“ in order to get ”ו“ the conjunction ”כחו “ second word Pet., L, NY 2696; the copyist of P has misread the text of his model, eliminating by [ידוע homoioteleuton several lines, including this passage.
260 Tony Lévy and Bernard Vitrac first, omitting “the power of” which was in his copytext. Paying more attention to the third and fourth times, he noticed that his copytext had the complex formula that should be simplified, whence the overbar on “the power of” to indicate that it should be deleted (see above, nn. 155, 157). This is only speculation on our part, but it suggests that Pet. 343/4 could thus be the sub-archetype of the family with the “simplified” formula. However, other variants do not fit with this surmise or would require some contamination between manuscripts:
• Pet. 343/4, Pet. and P omit the word “sages” from the phrase “ in the ,(see above, n. 60) (בספרי החכמים הקדמונים( ”books of the ancient sages whereas this word appears in L. • Pet. probably spaced out the text (as did NY) before inserting the 8 that is missing in Metrica I.25, whereas the blank space has vanished in Pet. 343/4, B, Pet. and L (see above, end of the fourth encounter).158 • Pet. and P have common mistakes: • They both omit “prismatic” at the end of the translation of Stereometrica II.57.3 (see above, n. 110). 1 • They both “correct” ∕6 to ⅓ at the end of Stereometrica II.60.1 (see above, nn. 115–116). • But we cannot prove that P was copied from Pet. or that both made use of the same copytext. Readers who consult our notes (see also above, nn. 153, 157) will readily agree that the copyist of the Paris codex was rather absent-minded. The text provides many confirmations of this.
That the history of Komtino’s text is more complex than indicated by the manuscripts we consulted is also clear from a very specific variant we found in L. In this codex, the passage designated (vi) above is completely changed by the random insertion of short statements. These are clearly lemmata (repeating parts of the text) and meta-textual remarks. It is most probable that these two elements were marginal glosses in the copytext of L. The scribe interpolated them into the text without understanding that they belonged to two different textual registers: the main text, and a marginal commentary comprising lemmata accompanied by
158 We suppose that the elimination in both B and Pet. 343/4 and P of the blank space indicating a lacuna in Metrica I.25 in S, NY, and Pet. (see above, end of the fourth encounter) took place independently of one another, by scribes who did not understand the reason for it.
261 Hero of Alexandria and Mordekhai Komtino very critical remarks (e.g., “thus you have multiplied the triangles by themselves”; “and you would not have been mistaken”; “this is false and impossible”). The author of these glosses (in the copytext of L) was disturbed by the idea of multiplying a rectangle by itself or a triangle by itself; he thought this was impossible and an error to be avoided. The presence, earlier and later, of other “four-dimensional” entities (dunamodunamis, the product of two squares) does not seem to have bothered him, perhaps because he took them for arithmetic entities, whereas here the terminology is clearly geometric. We observe that this corrupt sequence is found in almost identical form in NY 2696. On the other hand, because these supposed marginal glosses are not found in any other surviving manuscripts that antedate L (namely NY, Pet. 343/4, Pet., B, and P), we infer there was some other manuscript unknown to us, that contained these marginalia.
* The authors would like to thank the research units (AnHiMa and SPHERE at CNRS in Paris) that supported their work. Thanks also to the Sidney M. Edelstein Centre for the History and Philosophy of Science, Technology and Medicine in Jerusalem, which provided Tony Lévy with a senior visiting fellowship in 2015, making it possible for him to study all of the manuscript sources of Komtino’s work. Translated from French by Lenn Shramm (Jerusalem). We are very grateful to Mr. Schramm for his expert translation.
262