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The article presents the edition and a detailed discussion of an anonymous treatise of elementary arithmetic that served as the source of the so-called Letter to Khatzykes authored by the Byzantine scholar Nicholas Artabasdos Rhabdas. An updated survey of the extant evidence about the logistic treatises composed in the Nicaean period and in the early Palaiologan era, and a discussion of the prima facie surprisingly widespread phenomenon of appropriation of scientific treatises written by other contemporaries in late Byzantine times will also be provided. This article presents the edition of an anonymous treatise of elementary arithmetic (henceforth called “Anonymus B”) contained, in slightly incomplete form, in ff. 171r–186v of the manuscript Città del Vaticano, Biblioteca Apostolica Vaticana, Barberinianus gr. 4 (Diktyon 64552), to be dated to the beginning of the 14th century. The most important point of our study, however, does not lie in assessing Anonymus B in its own terms, but in showing that it served as the source of the so-called Letter to Khatzykes1 authored by the Byzantine scholar Nicholas Artabasdos Rhabdas2. That the First Letter and the anonymous treatise are very closely connected is obvious from their verbatim agreeing over large stretches of text; two series of facts decisively support our stronger claim about their filiation. First, the Barberini codex, produced within the circle of Maximos Planudes’ (d. ca. 1305) pupils, dates at the latest to the period of Rhabdas’ early activity. Second—and crucially, on account of the fact that the paleographical record does not completely settle the issue of priority—the involved variant readings, starting from the very titles of the two texts, strongly corroborate the hypothesis that Anonymus B is the source of Rhabdas’ First Letter, and not the inverse. As it also happens that an anonymous arithmetical treatise dated 1252 (henceforth “Anonymus 1252”) underwent a similar treatment in the hands of Planudes himself, resulting in his celebrated Great Calculation According to the Indians, we shall provide a revised outline of the extant evidence about the logistic treatises redacted in the Nicaean period (1204–61) and in the early Palaiologan era; we shall also argue, on grounds of style and contents, that Anonymus B and Anonymus 1252 were not composed by the same author. Consequently, we shall discuss the prima facie surprisingly widespread phenomenon, in late Byzantine times, of appropriation of scientific treatises written by other Byzantine scholars. The structure of the article is as follows. We first survey the evidence about Rhabdas’ scientific career; we then briefly describe Barb. gr. 4. The edition of Anonymus B, including the accompanying tables, is followed by an analysis of the textual differences with respect to Rhabdas’ First Letter, and by an assessment of the flourishing of logistic treatises in the middle of the 13th century. In the last * We are grateful to F. Valerio for a preliminary check of Vat. gr. 1481, to O. Delouis for pointing out a relevant text to us. This manuscript, Barb. gr. 4, and Chis. R.IV.20 have been collated at the Biblioteca Apostolica Vaticana on June 6–8 and July 10–12, 2018. FA and IPM have been supported by the Research project “The Byzantine Author (II)” (MICINN, FFI201565118-C2-2-P). DM’s contribution was written as part of the project UMO-2015/19/P/HS2/02739, supported by the National Science Centre, Poland; this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 665778. 1 Henceforth called First Letter. We shall see that there are two “letters” of arithmetical content authored by Rhabdas. 2 Rhabdas was born in Smyrna and was active in Constantinople about 1320–40; see PLP, no. 1437; ODB, 1786–1787; and the discussion in the next section. F. Acerbi – D. Manolova – I. Pérez Martín 2 section, we present factual evidence and some considerations on the issue of “scientific plagiarism” in Palaiologan Byzantium. NICHOLAS RHABDAS, LIFE AND WORKS Enough of Nicholas Rhabdas’ scholarly production has been preserved for us to acknowledge his expertise in the mathematical sciences and especially his significant contribution to the domain of Byzantine logistic3. This is a branch of arithmetic in which a unit can be divided and that deals with counting numbers and with calculations on them4. Logistic developed greatly in Late Antiquity as a support to mathematical astronomy, and retained this role in Byzantine times5. As we shall see, Rhabdas was also engaged in astronomical matters and, in addition, composed a grammatical treatise for his son Paul Artabasdos6. 3 His production, however, was assessed in a way that is paradigmatic of a generalized dismissive attitude to Byzantine science; P. Tannery, Manuel Moschopoulos et Nicolas Rhabdas. Bulletin des Sciences mathématiques, 2e série, 8 (1884) 263–277, repr. Id., Mémoires Scientifiques IV. Toulouse–Paris 1920, 1–19: 15, in fact, passed the following judgment on Rhabdas’ writings, whose edition he nevertheless published two years later: “L’intérêt de ses écrits est surtout de montrer jusqu’où étaient tombés les héritiers dégénérés du nom hellène, ceux-là même qui avaient alors Diophante entre leurs mains”. 4 According to the 6th-century Neoplatonic commentator Eutocius, dividing the unit does not pertain to arithmetic but to logistic (J. L. Heiberg (ed.), Archimedis opera omnia cum commentariis Eutocii. I–III. Lipsiae 1910–15, III 120.28–30: ὥστ’ ἐπ’ ἐκείνων [scil. superparticular and superpartient ratios] διαιρετέον τὴν μονάδα, ὃ εἰ καὶ μὴ κατὰ τὸ προσῆκον τῇ ἀριθμητικῇ ἀλλὰ τῇ λογιστικῇ τυγχάνει “so that, for them one has to divide the unit, even if this does not happen to fit to arithmetic, but to logistic”). An earlier definition of logistic—which can almost certainly be ascribed to Geminus (a 1st-century BCE mathematically-minded philosopher, maybe a pupil of Posidonius)—does not allow dividing the unit. This definition can be found in pseudo-Hero, Def. 135.5–6 (J. L. Heiberg – L. Nix – W. Schmidt – H. Schöne (eds.), Heronis Alexandrini opera quae supersunt omnia. I–V. Lipsiae 1899–1914, IV 98.12–100.3), and is also preserved, through a different line of tradition, as a scholium to Plato, Chrm. 165e6 (Scholium 27 in D. Cufalo, Scholia Graeca in Platonem. I. Scholia ad dialogos tetralogiarum I–VII continens [Pleiadi 5.1]. Roma 2007, 173). It is possible that the domain of logistic was expanded to include fractional parts as a consequence of the adoption of the sexagesimal system in Greek mathematical astronomy. In fact, logistic developed greatly in Late Antiquity as a support to mathematical astronomy, and also played this same role in the Byzantine period. The first known treatise of this kind is included in the Prolegomena to the Almagest, and amounts to the (non-redacted) lecture notes of a course held in the circle of the Neoplatonic philosopher Ammonios. This treatise is a computational primer to the Almagest: a tightly organized “handbook of logistic” featuring as its main themes an introduction to the sexagesimal system, a description of computational algorithms for multiplication, division, and extraction of an approximate square root, a presentation of interpolation techniques, and an exposition about compounded ratios and removal of a ratio from a ratio. According to the anonymous author, no comprehensive previous exposition of this kind existed— and in fact no such one has been transmitted to us. See also note 125 below. The best introduction to Greek logistic is still K. Vogel, Beiträge zur griechischen Logistik. Erster Teil (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Abteilung). Munich 1936, 357–472. 5 Cf. the explicit statement opening Anonymus 1252: A. Allard, Le premier traité byzantin de calcul indien: classement des manuscrits et édition critique du texte. Revue d’Histoire des Textes 7 (1977) 57–107: 80.2–4, and, in a smoother formulation, Planudes’ Great Calculation: A. Allard (ed.), Maxime Planude, Le grand calcul selon les Indiens. Louvain-la-Neuve 1981, 27.1–5. Despite its title (and the author’s statement similar to that of Planudes: P. Carelos [ed.], Βαρλαὰμ τοῦ Καλαβροῦ, Λογιστική, Barlaam von Seminara, Logistiké [Corpus philosophorum Medii Ævi. Philosophi byzantini 8]. Athens–Paris– Brussels 1996: 1.10–26), Barlaam’ Logistic is not a writing of logistic, but a fully-fledged treatise of theoretical arithmetic formulated in an impeccable demonstrative style. Barlaam (PLP, no. 2284), undisputably the Byzantine scholar best versed in mathematical matters and a major actor in the hesychastic controversy, died in 1348. 6 See PLP, no. 1438. The unpublished grammatical synopsis addressed to Paul is preserved in the miscellaneous ms. Paris, Bibliothèque nationale de France, gr. 2650 (Diktyon 52285), ff. 147r–150v. The copying is dated December 6, a.m. 6836 [= 1427] (f. 204v), certainly referring to ff. 201r–204v and possibly also to ff. 153r–167v and 168v–198v, penned by the same hand. However, the script of Rhabdas’ synopsis (located in quire no. ιθʹ, a ternion closed by the blank ff. 151r–152r) seems earlier, perhaps dating back to the middle – third quarter of the 14th century. The synopsis is presented as a grammatical compendium whose aim is expounding the appropriate use of words, in order to avoid barbarisms and solecisms. The exposition is based on analytical divisions of the main grammatical issues, treated by means of μικρούς τινας ὑπομνηματισμούς “some short notes”. It starts from letters (γράμματα and στοιχεῖα) and goes on dealing with syllables and words insofar as they are 3 The Source of Nicholas Rhabdas’ Letter to Khatzykes Much less is known about Rhabdas’ life, education, personal and professional networks. Until recently, the only temporal clue was provided by the fact that, i

The Source of Nicholas Rhabdas’ Letter to Khatzykes: An Anonymous Arithmetical Treatise in Vat. Barb. gr. 4