AES EXCURRENS and the ABACUS* in the Text Distributio Partium (DP) by Lucius Volusius Maecianus (Mae- Cianus, for Short)1, Which
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AESEXCURRENS AND THE ABACUS* Abstract: This paper seeks to shed some new light on the function of the abacus (counting board) as an instrument for financial computa- tions in Roman times. The likely layout of the abacus is presented for the Roman monetary notations for aesexcurrens (small change) in the as-, sestertius- and denarius-based accounting systems mentioned in L. Volusius Maecianus’ Distributio Partium. It is shown that the Roman abacus possessed the flexibility needed for handling both the common Roman duodecimal fractions and the less common decimal fractions, i.e. the libella and its subdivisions. In turn, the very design of the abacus seems to have driven the later Roman notations for small change. INTRODUCTION, OBJECTIVES, ORGANISATION In the text DistributioPartium (DP) by Lucius Volusius Maecianus (Mae- cianus, for short)1, which is one of the few primary sources for Roman metrology, reference is made to the different notations that the Romans supposedly used to record sums of money. Monetary amounts were regis- tered in terms of multiples and fractions of the accounting units in use: at the time a clear distinction was made between as-, sestertius- and denarius- based monetary notations and types of accounting, known as the ratio aeraria, ratiosestertiaria (or ratioadsestertium) and ratiodenariaria (or * The author wishes to express his gratitude to Prof. Dr. S. Cuomo, Department of History, Classics and Archaeology, Birkbeck College, University of London, U.K., to Prof. Dr. J.P. Hogendijk, Mathematical Institute, University of Utrecht, The Netherlands and to Dr. F.A.J. Hoogendijk, Papyrological Institute, Leiden University, The Netherlands, for reading early drafts of this paper and their valuable suggestions for improvement. The professional comments by the (anonymous) reviewers of AncientSociety were also very helpful and highly appreciated. The author further wants to thank em. Prof. Dr. H. Grassl, Alte Geschichte und Altertumskunde, Salzburg University, Austria, for kindly making a draft paper on ‘Römische Kleingeldrechnungen’ available, prior to its final publication. Of course any errors and omissions in the final version of this article are the sole respon- sibility of the author. 1 Lucius Volusius Maecianus (c. 110-166 CE) was a respected jurist under the emperors Hadrian, Antoninus Pius and Marcus Aurelius, see Liebs (1997, 2010). For further details of his military and civil careers, see the electronic portal PIR2 (2015), entry V 973. Avail- able text editions of Maecianus’ DistributioPartium (DP) include those by Mommsen (1853) 281-295, Hultsch (1864) 61-71 and Huschke e.a. (1908) 409-418; in the current paper Mommsen’s edition of the DP has been used throughout. AncientSociety 48, 115-145. doi: 10.2143/AS.48.0.3285199 © 2018 by Ancient Society. All rights reserved. 116 F. DE BREE ratioaddenarium), respectively.2 In his DP, Maecianus explains their use to his emperor to be, Marcus Aurelius, and points to a remarkable diversity in the ‘official’ notations3 for aesexcurrens (small change), i.e. for money with a value less than that of the accounting unit. Depending on the accounting system used, small change was apparently recorded either in terms of the Roman duodecimal fractions (multiples of the uncia = 1/12 as and smaller sub-units) or in terms of the archaic — and much less common — decimal fractions libella =1/10, sembella = singula = 1/20, and ter(r) uncius = 1/40. As suggested by Maecianus, most likely the libella was originally, at the time of the Early Roman Republic, a silver coin and the counterpart of the bronze as, worth 1/10 of the early denarius, but later on survived merely as a fraction equal to 1/10. The above exotic decimal fractions are also mentioned by earlier clas- sical authors, though very sporadically.4 As to the secondary literature on this topic, it appears that the various types of Roman monetary notations were discussed in the past, notably in a number of Austrian and German contributions dating from 30 to 100 years ago.5 In particular, Alfred Nagl’s 1914 paper, entitled ‘Die Rechen- tafel der Alten’, is to be seen as a key reference, providing an early and comprehensive discussion of the DP of Maecianus and hinting also at a plausible connection between the build-up of some of the above mone- tary notations for small change and the layout of the Roman abacus (counting board).6 The most recent literature generally seems to virtually ignore these subjects.7 Yet, the actual existence of the notations for small change, mentioned in Maecianus’ DP, was unequivocally confirmed in a 2 Cicero, ProPublioQuinctio17, mentions the ratioaeraria; Maecianus (in DP [48- 63, 64-73]) focuses mainly on the ratiosestertiariaand denariaria. 3 The qualification ‘official’ is used here under the assumption that Maecianus would share with his emperor mainly those monetary notations that had some official status. 4 For explicit literary references made by classical authors to the libella and its subdi- visions, see below, notes 22-23. 5 Compare Friedlein (1869), Beigel (1904), Nagl (1914, 1918), Röhle (1987). 6 See Nagl (1914), notably ch. III, p. 66-77. For a brief overview of that work, see also Nagl (1918), consisting of three additional articles in the RE, entitled ‘Abacus’, ‘Aes excurrens’ and ‘Kleingeldrechnung’. 7 In a detailed paper by Cuomo (2007) on the DP of Maecianus, various Roman met- rological characteristics were discussed extensively; however, little attention was given to the remarkable diversity in the monetary notations mentioned by Maecianus for small change (the main subject of the current paper). The same applies to the dissertation of LaGroue (2014), geared specifically to Roman accounting and auditing. Recent work which does deal with Latin notations for small change includes the studies by Grassl (2014, 2015 and 2018) on the lead and wooden tablets from a variety of Roman provinces, AESEXCURRENSAND THE ABACUS 117 recent exploratory/inventory study by the author, based on focused searches in the nowadays readily available literary, epigraphical and papyrological source material; in the same study also an assessment of the approximate chronology of the different types of Roman accounting could be made.8 The current study is to be seen as a companion to that exploratory/inventory study, specifically devoted to the use of the Roman abacus for handling small change. When considering the professional practice of accounting, the abacus comes into the picture as an early instrument suitable for elementary arithmetic operations, including financial computations. In fact, the sparse extant physical examples of the Roman abacus show symbols that have both a numeric and a purely monetary connotation, thus hinting at its use as an accounting tool. Also, the very wording aesexcurrens directly refers to the physical handling of small change on the abacus and cer- tainly provides another strong pointer to the abacus having been used for accounting purposes.9 Although the operation of the classical abacus has been extensively treated in the literature, the descriptions about its use for the specific handling of small change generally seem underdeveloped.10 For this reason, the majority of which were (re-)interpreted in terms of the common duodecimal notations pertaining to the denarius type of accounting. 8 See De Bree (2019), forthcoming. 9 In view of the definition of small change as money smaller than (i.e. being a frac- tion of) the accounting unit, it was physically handled on the right-hand side of the Roman abacus. As will be discussed in more detail in the main text of this article, frac- tions were manipulated by up- and downshifting the beads in the various slots grooved in the abacus’ surface. Obviously, this had to be done without letting these beads run over and drop from the edges of the counting board, as this would make them literally aes excurrens (‘money running over’), which is the wording used by Maecianus in DP [63]. Similar expressions for small change are found elsewhere: it is, for example, referred to as et quod excurrit (‘what runs over’) on the inscription CIL VI 33840, 1 (Rome, 25 July, AD 227) = EDCS-24100613, which is further discussed by Gordon (1983) 161 and Röhle (1987). At this point, it is to be noted that in practice an easy way to prevent the beads from running over was to design the abacus with raised edges: compare Schärlig (2004) 68, discussing an engraved abacus with such edges, actually found in Trier, Germany. 10 For a description of the supposed use of the various extant classical abaci, the main early references are Friedlein (1869) and, again, Nagl (1914), who both mention the nota- tions of Maecianus for small change, but do not give all the operational details that are provided in the current article. More recent literature about the abacus is found in Pullan (1968), Fellmann (1983), Ifrah (1985), Schärlig (2001, 2003, 2004, 2006) and Stephenson (2013): none of these refer to Maecianus’ DP. Also worth mentioning, in this connection, are the works of Turner (1951) and Maher & Makowski (2001), who discuss how various Roman authors seem to have dealt with 118 F. DE BREE it was felt that a more complete overview, appreciating the fundamental and rich functionality of the Roman abacus, was desirable, the more so now that a fully recovered Roman abacus (found in Aosta, Italy, discussed below) is available. The main question that will be addressed in this paper is how precisely the Roman counting board could have handled the different types of Roman monetary notations, notably the different fractions needed to represent small change. To that end, first a brief description will be presented of the well- established and generally accepted functionality of the abacus for operations with whole numbers and the common well-known duodecimal fractions.