Muscovite Arithmetic in Seventeenth-Century Russian Civilization: Is It Not Time to Discard the âBackwardnessâ Label?*
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Muscovite civilization utilized Byzantine-Greek alphanumerals for its mathematical symbols. Occasionally derided by historians for being retrograde in comparison to the Hindu-Arabic numerals sixteenth- and seventeenth-century Europe adopted, Muscovyâs alphanumerals were versatile and suitably contoured to perform a variety of computational tasks. Muscovite alphanumerals were an integral part of early Moderen Russiaâs administrative culture, and played a prominent role in fostering the experiential knowledge underlying the educational achievements of the Imperial Period. Though they lacked the zero and the decimal, Muscovites still had a reasonable grasp of the base-ten system, and comprehended well basic arithmetical skills and relationship properties, less so equational ones. The Russians developed complex abaci well suited for commercial transactions, large-scale construction, military inventories and payrolls, and the land registry, to name a few. These instruments manipulated an extensive variety of weights, measures, linear distances, area dimensions, volume measurements, and currency. Muscovite arithmetic was a prominent factor assisting in the advancement of critical thinking skills in 1600âs Russia. Nonetheless, as the seventeenth century wore on, sociological, educational or pedagogical, military scientific, administrative, and cultural arguments or interactive phenomena came to bear and increasingly found the Muscovite algorithmic symbols wanting. In 1699 the government decreed that Hindu-Arabic numerals henceforth were to be used in official documents throughout the country. Directly and indirectly, the complex thought processes bound up when operating with Muscovite alphanumerals were one impetus for the further unfolding of Russian civilization after 1700.
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1)
âGeorges Florovsky, âThe Problem of Old Russian Culture,â Slavic Review 21, no. 1 (March 1962): 1-15 and ibid., in The Structure of Russian History. Interpretive Essays, ed., Michael Cherniavsky (New York: Random House, 1970), 126-39.
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2)
âFlorovsky, âThe Problem of Old Russian Culture,â 133-36.
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3)
âSee Robert Romanchuk, Byzantine Hermeneutics and Pedagogy in the Russian North: Monks and Masters at the Kirillo-Belozerskii Monastery, 1397-1501 (Toronto: University of Toronto Press, 2007).
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4)
âRichard Hellie, âLate Medieval and Early Modern Russian Civilization and Modern Neuroscience,â in Culture and Identity in Muscovy, 1359-1584, eds., A.M. Kleimola, G.D. Lenhoff (Moscow: ITZ-Garant, 1997), 146-55; id., âThe Russian Smoky Hut and Its Probable Health Consequences,â Russian History, 28, nos. 1-4 (Spring, Winter, Fall, Winter 2001): 171-84.
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5)
âE.g., Peter B. Brown, âPeering into a Muscovite Turf-War (How Do We Even Know Itâs There?): Boyar Miloslavskii and the Auditing Chancellery,â Russian History 25, nos. 1-2 (Spring-Summer 1998): 141-53; id., âSalaries and Economic Survival: the Service Land Chancellery Clerks of Seventeenth-Century Russia,â Jahrbücher für Geschichte Osteuropas 51 (2003): 32-67; id., âBureaucratic Administration in Seventeenth-Century Russia,â in Modernization of Muscovy, eds., Jarmo Kotilaine, Marshall Poe (London and New York: RoutledgeCurzon, 2004), 57-78; id., âHow Muscovy Governed: Seventeenth-Century Russian Central Administration,â Russian History 36, no. 4 (2009): 503, 520. Concentrating more upon the late fifteenth and early sixteenth centuries, Marshall Poe, through detailed examination of a wide variety of documents, argues how the replication of this paperwork sharpened administrative appetites and led to the steady obtrusion of these instruments and the practices embodied in them throughout the realm. See Marshall Poe, âThe Military Revolution, Administrative Development, and Cultural Change in Early Modern Russia,â Journal of Early Modern History 2, no. 3 (August 1998), 247-73; id., âMuscovite Personnel Records, 1475-1550: New Light on the Early Evolution of Russian Bureaucracy,â Jahrbücher für Geschichte Osteuropas 45, no. 3 (1997): 361-77; id., âElite Service Registry in Muscovy, 1500-1700,â Russian History 21, no. 3 (Fall 1994): 253-83.
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9)
âRichard Hellie, Slavery in Russia 1450-1725 (Chicago: University of Chicago, 1982), 237 makes a similar point.
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âM.B. Parkes, Scribes, Scripts, and Readers. Studies in the Communication, Presentation, and Dissemination of Medieval Texts (London; Rio Grande, Ohio: the Hambledon Press, 1991), 280.
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âGary J. Marker, âPrimers and Literacy in Muscovy: A Taxonomic Investigation,â The Russian Review 48, no. 1 (January 1989): 1-19.
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13)
âNigel Wheale, Writing and Society. Literacy, Print, and Politics in Britain 1590-1660 (London, New York: Routledge, 1999), 2. These are aggregate figures inclusive of both genders and all social strata (ibid.).
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15)
âAmitai Etzioni, A Comparative Analysis of Complex Organizations. On Power, Involvement, and Their Correlates (New York: the Free Press, 1971), 203-04.
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âR.A. Simonov, Matematicheskaia i kalendarno-astronomicheskaia myslâ Drevnei Rusi. Po dannym srednevekovoi knizhnoi kulâtury (Moscow: Nauka, 2007), 37-40, 46, 51-52, 61-65, 68, 70, 76-79, 97, 100, 108-17. The three Russkaia Pravda redactions (Short, Expanded, Abbreviated) from Iaroslavâs reign (1019-54) and after frequently introduce numbers in regards to damages, fines, court fees, and allowances. See Daniel H. Kaiser, trans., ed., The Laws of RusââTenth to Fifteenth Centuries (Salt Lake City, Utah: Charles Schlacks, Jr., Publisher, 1992), 15-33, 35-39. One can pick at random any Russian chronicle from the series Polnoe sobranie russkikh letopisei and find Byzantine-Slavonic numerals, e.g., in the first volume: Lavrentâevskaia letopisâ (Moscow: Iazyki russkoi kulâtury, 1997), passim; in the third volume: Novgorodskaia pervaia letopisâ starshego i mladshego izvodov (Moscow: Iazyki russkoi kulâtury, 2000), passim; in the ninth volume: Letopisnyi sbornik, imenuemyi Patriarshei ili Nikonovskoi letopisâiu (Moscow: Iazyki russkoi kulâtury, 2000), passim. Since the early 1950âs, the on-going, multi-volume series (Novgorodskie gramoty na bereste) of the excavated birchbark charters from the Novgorod region contain copious amounts of quantitative data inscribed in letter-numerals.
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24)
âBrown, âBureaucratic Administration in Seventeenth-Century Russia,â 77; N.M. Demidova, Sluzhilaia biurokratiia v Rossii xvii v. i ee rolâ v formirovanii absoliutizma (Moscow: Nauka, 1988), 43; Borivoj Plavsic, âSeventeenth-Century Chanceries and Their Staffs,â in Russian Officialdom. The Bureaucratization of Russian Society From the Seventeenth to the Twentieth Century, eds., Walter McKenzie Pintner, Don Karl Rowney (Chapel Hill: University of North Carolina Press, 1980), 29; S.K. Bogoiavlenskii, âPrikaznye dâiaki xvii v.,â Istoricheskie zapiski, 1 (1937): 220-39.
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28)
âEves, Introduction to the History of Mathematics, 14; H.L. Resnikoff, R.O. Wells, Jr., Mathematics in Civilization (New York: Holt, Rinehart, and Winston, Inc., 1973), 24-25; Tobias Dantzig, Number. The Language of Science, ed., Joseph Mazur (New York: Plume, 2007), 303-04; Tobias Dantzig, Mathematics in Ancient Greece (New York: Charles Scribnerâs Sons, 1955; reprint, 2006) provides widely-encompassing and detailed analyses of Hellenic and Hellenistic theoretical and applied mathematics.
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30)
âV.A. Petrova, Paleograficheskii alâbom. Uchebnyi sbornik snimkov s rukopisei russkikh dokumentov xiii-xviii vekov (Leningrad: LGU, 1968), 19; Gavin Betts, Latin (Dunton Green, England: Hodder and Stoughton, 1986), 80, 276. The travel accounts by Sigmund von Herberstein, Richard Chancellor, Anthony Jenkinson, Sir Thomas Randolph, George Turberville, Giles Fletcher, Sir Jerome Horsey, Heinrich von Staden, Antonio Possevino, and Adam Olearius are unrewarding for information on the Muscovitesâ usage of letter-numerals, though Possevino does examine how the Russians dated from the Flood (Antonio Possevino, Istoricheskie sochineniia o Rossii xvi v., trans., L.N. Godovikovaia [Moscow: MGU, 1983], 208).
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31)
âSee L.V. Cherepnin, Russkaia paleografiia (Moscow: Gosizdatpolitlit, 1956), 373.
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34)
âBoyer, Merzbach, A History of Mathematics, 240-41; Edmund Fryde, The Early Palaeologan Renaissance (1261-1360) (Leiden, Boston, Cologne: Brill, 2000), 338-41.
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36)
âZ.V. Dmitrieva, âMetrologiia,â in Spetsialânye istoricheskie distsipliny. Uchebnoe posobie, ed., M.M. Krom (St. Petersburg: Dmitrii Bulanin, 2003), 476-506; V.B. Kobrin, G.A. Leontâeva, and P.A. Shorin, Vspomogatelânye istoricheskie distsipliny (Moscow: Prosveshchenie, 1984), 65-72; ibid. (Moscow: Vlados, 2000), 263-70.
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38)
âSimonov, Matematicheskaia i kalendarno-astronomicheskaia myslâ, 59-60. One of the reviewers noted that âBoth Byzantine and Latin systems are additive, and neither becomes easier once other arithmetical operations are required. In both cases they required the use of tools (e.g., abacus or tables).â
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39)
âMario Livio, âWhy Math Works,â Scientific American, August 2011, 82-83.
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41)
âLakoff, Nuñez, Where Mathematics Comes From, 19, 64, 84; Boyer, Merzbach, A History of Mathematics, 239. See the discussion in John Cottingham, ed., The Cambridge Companion to Descartes (New York: Cambridge University Press, 1992), 101-02.
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46)
âPeter J. Bentley, The Book of Numbers. The Secret of Numbers and How They Changed the World (Buffalo, New York: Firefly Books, Inc., 2008), 167-69. For more on the Babylonians see H.L. Resnikoff and R.O. Wells, Jr., Mathematics in Civilization (New York: Holt, Rinehart, and Winston, Inc., 1973), 25.
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47)
âW.F. Ryan, âScientific Instruments in Russia from the Middle Ages to Peter the Great,â Annals of Science. The History of Science and Technology from the Thirteenth Century, 48, no. 4 (July 1991), 373.
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49)
âHelmuth Gericke, âDie Zeit von 1500 bis 1637,â Mathematik im Abendland. Von den römischen Feldmessern bis zu Descartes (Berlin, Heidelberg, New York: Springer-Verlag, 1980), 225-96.
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51)
âV.A. Petrova, comp., Paleograficheskii alâbom. Uchebnyi sbornik snimkov s rukopisei russkikh dokumentov xiii-xviii vv. (Leningrad: Izdatelâstvo LGU, 1968), 19, 33, 41, 49; M.N. Tikhomirov, A.V. Muravâev, Russkaia paleografiia (Moscow: Vysshaia shkola, 1966), 4-5; L.V. Cherepnin, Russkaia khronologiia (Moscow: Glavnoe arkhivnoe upravlenie NKVD SSSR, 1944), 20-24; Grigore NandriÅ, Handbook of Old Church Slavonic, 2 pts. corrected ed. (London: Athlone Press, 1965), 1: 3; Simonov, Matematicheskaia i kalendarno-astronomicheskaia myslâ, 60. I use the Middle Russian names and not the slightly different Old Church Slavonic names for the Muscovite letter-numerals.
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55)
âJohn Allen Paulos, Innumeracy. Mathematical Illiteracy and Its Consequences (New York: Hill and Wang, 2001), 11, 28, 44, 108-10, 141-42; Arthur Benjamin and Michael Shermer, Secrets of Mental Math. The Mathemagiciansâs Guide to Lightning Calculation and Amazing Math Tricks (New York: Three Rivers Press, 2006), 203-04.
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56)
âW.F. Ryan, âJohn Tradescantâs Russian Abacus,â Oxford Slavonic Papers, n.s., 5 (1972): 87; id., âScientific Instruments in Russia,â 373. For an introduction to that appliance, see Jesse Dilson, The Abacus. The Worldâs First Computing System: Where It Comes From, How It Works, and How to Use It to Perform Mathematical Feats Great and Small (New York: St. Martinâs Griffin, 1968), 33-55 and Boyer, Merzbach, A History of Mathematics, 225-27.
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59)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 332, 336.
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66)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 336-37.
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70)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 315-17. Variants are depicted in ibid., 306, 308. âThe origins of the doshchanoi or doshchatoi schet remain obscure. Western European counting boards may have parentedâ (reviewerâs comment). One pound is .454 kg (Pocket Oxford Dictionary, 456).
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71)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 328, 347.
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72)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 305-21, 328-29; W.F. Ryan, âAbacus (Western),â in Instruments of Science. An Historical Encyclopedia, eds., Robert Bud, Deborah Jean Warner (New York, London: Garland Publishing, Inc., 1998), 7; A.P. Iushkevich, Istoriia matematiki v Rossii do 1917 goda (Moscow: Nauka, 1968), 27-30. These citations give an understanding of how in Russia the schet or simple counting board evolved into the doshchanoi schet. One acre is 4,840 square yards or 4,046.9 square meters (Pocket Oxford Dictionary, 9).
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73)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 308, 311-16.
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74)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 308-09, 312-15, 335-36. Possibly there were yet other configurations for the seventeenth-century complex calculationsâ board (ibid., 319).
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75)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 318.
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77)
âIushkevich, Istoriia matematiki v Rossii, 28; Ryan, âScientific Instruments in Russia,â 37.
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78)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 321-27, 331-33.
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79)
âIushkevich, Istoriia matematiki v Rossii do 1917 goda, 15-16, 28.
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80)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 322-27.
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82)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 322-27.
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86)
âBoyer, Merzbach, History of Mathematics, 69; Spasskii, âProiskhozhdenie i istoriia russkikh schetov,â 338-46. For an American (like the author) or for a Muscovite, neither of whom was brought up on the metric system, routinely dividing and sub-dividing the units in the English and Muscovite measurement systems by one-half and one-third is easy and âcomes naturallyâ and seems un-foreign and less-contrived than having to learn how to calculate in a different numeral- or operational-based system. The penetration of English measurements into American commercial practice is certainly one reason for the longevity of non-metric measurement in the U.S., and something similar to this took place in Muscovy vis à vis Byzantine letter-numerals. Without question various self-reinforcing cultural influences came/come into play for the persistence of Byzantine-Slavonic letter-numerals and English measurements in Muscovy and in the U.S. respectively.
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87)
âSee Lakoff, Nuñez, Where Mathematics Comes From, 94-95. Curiously, the later Muscovites did not use the few shorthand symbols the Byzantines developed for fractions (Oxford Dictionary of Byzantium, 1: 1501).
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90)
âVeselovskii, Soshnoe pisâmo, 1: 44-47, 87-88, 93-94. Nonetheless, the nagging problem of land measurement unitsâ non-specificity remained (ibid., 2: 408-12). For measurements of average numbers of chetverti for urban households (posadskie liudi), see ibid., 2:263-337).
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91)
âSpasskii, âProiskhozhdenie i istoriia russkikh schetov,â 373-74; P.N. Miliukov, Gosudarstvennoe khoziaistvo Rossii v pervoi chetverti xviii veka, 2d ed. (St. Petersburg: Tipografiia M.M. Stasiulevicha, 1905), 42. The zhivushchaia chetâ (âthe inhabited chetâ) and zhivushchaia vytâ (âthe inhabited vytâ), introduced experimentally in the 1630âs into some monastery and military service estates, based themselves on number of households in a given land unit, not on land productivity as for the bolâshaia sokha and its fractionable units. Only with the household taxâs universalization in the late 1670âs, did the government, in an attempt to drum up more revenue, abandon the older and by now obsolete bolâshaia sokha. Using households as units had one advantage, namely they could not be fractionalized unlike amounts of land (Dmitrieva, âMetrologiia,â 489-90; N.A. Rozhkov, Russkaia istoriia v sravnitelâno-istoricheskom osveshchenii. Osnovy sotsialânoi dinamiki, 2d ed. [Petrograd, Moscow: Kniga, early 1920âs], 4, pt. 2: 88; N.V. Ustiugov, âFinansy,â 418-19).
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92)
âVeselovskii, Soshnoe pisâmo, 2: 393; N.V. Ustiugov, âFinansy,â in Ocherki istorii SSSR. Period feodalizma xvii v., eds., A.A. Novoselâskii, N.V. Ustiugov (Moscow: AN SSSR, 1955), 420; G.V. Alferova, âMatematicheskie osnovy russkogo gradostroitelâstva xvi-xvii vv.,â in Estestvennonauchnye znaniia v Drevnei Rusi, ed., R.A. Simonov (Moscow: Nauka, 1980), 131; I.V. Levochkin, âOb izmerenii selâskokhoziaistvennykh ugodii v tsentralânoi Rossii xvi-xvii vv.,â in Estestvennonauchnye znaniia v Drevnei Rusi, 134-35; Dmitrieva, âMetrologiia,â 487-91; M.M. Krom, Spetsialânye istoricheskie distsipliny. Uchebnoe posobie (St. Petersburg: Dmitrii Bulanin, 2003), 487-88; Richard Hellie, Enserfment and Military Change in Muscovy (Chicago: University of Chicago Press, 1971), 125. Church and peasant privately-owned lands (âblackâ or taxable lands) received less advantageous scales: 900, 800, 700, and 600 chetverti, ranging from miserable to good quality for the former and 700, 600, and 500 for the latter (Dmitrieva, âMetrologiia,â 488). It should be obvious from the above how the government favored its service castes the most, by extracting less revenue from them, in comparison to the church, whose land tax assessment burden was in the middle, and the independent peasantsâ obligation, which was at the top. Today, six hundred (600) acres, just shy of one square mile, represents a substantial piece of land for a father and son, owning a family farm, to plow. Back then, the Russians, in the case of their service strata, would have had multiple serf and slave families to do the work.
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96)
âVeselovskii, Soshnoe pisâmo, 1: 58; Hellie, Economy and Material Culture, 648. For more such examples see Veselovskii, Soshnoe pisâmo, 1: 50, 72; 2: 432.
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97)
âK.S. Nosov, Russkie kreposti kontsa xv-xvii v. (St. Petersburg: Fakulâtet filologii i iskusstv Sankt-Peterburgskogo gosudarstvennogo universiteta, 2009), 38. Discussions on the military significance of Smolensk and its fortress, the construction of the Smolensk Wall, and the seventeenth-century conflicts fought over it are in Richard Hellie, Enserfment and Military Change in Muscovy (Chicago: University of Chicago Press, 1971); John L.H. Keep, Soldiers of the Tsar. Army and Society in Russia 1462-1874 (Oxford, New York: Oxford University Press, 1985); Robert I. Frost, The Northern Wars. War, State and Society in Northeastern Europe, 1558-1721 (Harlow, Great Britain: LongmanâPearson Education Limited, 2000); Carol B. Stevens, Russiaâs Wars of Emergence 1460-1730 (Harlow, England; London: Pearson Longman, 2007); Brian L. Davies, Warfare, State, and Society on the Black Sea Steppe, 1500-1700 (London, New York: Routledge, 2007).
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100)
âKostochkin, Fedor Konâ, 15-18, 35-38. The fact that the government constructed over half a hundred defensive fortifications in a twenty-year period suggests that the draconian edicts referred to in the preceding paragraph were not implemented fully.
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102)
âOrlovskii, Smolenskaia stena, 37; G.L. Kurzov, O kreposti staroi (Smolensk: Gody, 2003), 8-10. The 320,000 figure for both the barrels of lime and the piles is coincidental. One foot is 12â or 30.48 cm and one ton (American or short ton) is 2,000 pounds or 907.2 kg. One pood is 36.11 pounds or 16.38 kg (Pocket Oxford Dictionary, 312; Hellie, Economy and Material Culture of Russia, 648).
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103)
âPeter B. Brown, âMuscovite Government Bureaus,â Russian History 10, pt. 3 (1983): 298. One might expect that the central administration, supposedly in the avant-garde of information collection, would not have caved into recidivistic, Appanage Russian practices by authorizing the building of tall, relatively thin walls, plainly obsolete in the new era of cannons, and decreed instead the construction of the modern, low-slung, sharp-angled bastion fortresses. The Russians, for reasons unclear to the author, considered the latter unreliable. See Nosov, Russkie kreposti, 104-05. Both Konâ and the government carried on with the Italian tradition of using brick and de-emphasizing stone, for brick is significantly less time-consuming and less expensive to produce than cutting and transporting stone (ibid., 91-92).
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105)
âSee Valerie A. Kivelson, Cartographies of Tsardom: the Land and its Meanings in Seventeenth-Century Muscovy (Ithaca, New York: Cornell University Press, 2006). To read about how the Muscovites understood geometry and comprehended town-wall building techniques, see G.V. Alferova, âMatematicheskie osnovy russkogo gradostroitelâstva xvi-xvii vv.,â 109-33; id., âOrganizatsiia stroitelâstva gorodov v russkom gosudarstve xvi-xvii vekakh,â Voprosy istorii, 1977, no. 7: 50-66; id., âGosudarstvennaia sistema stroitelâstva gorodov i osvoenie novykh zemelâ v xvi-xvii vv. (na primere goroda Kozlova i ego uezda),â Arkhitekturnoe nasledstvo, 1979, no. 27: 1-16; id., âK voprosu o stroitelâstve gorodov Moskovskim gosudarstvom v xvi-xvii vv.,â Arkhitekturnoe nasledstvo, 1980, no. 28; A.A. Piletskii, âSistema razmerov i ikh otnoshenii v drevnerusskoi arkhitekture,â Estestvennonauchnye znaniia v Drevnei Rusi, 63-109. In that vein see also D.V. Liseitsev, âK voprosu o Gorodovom prikaze kontsa xvi-nachala xvii veka,â in Rossiiskaia realânostâ kontsa xvi-pervoi poloviny xix vv.: ekonomika, obshchestvennyi stroi, kulâtura. Sbornik statei k 80-letiiu Iu.A. Tikhonova (Moscow: RAN, 2007), 10-36. One yard is .914 meters (Oxford Pocket Dictionary, 975).
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108)
âHellie, âThe Costs of Muscovite Military Defense and Expansion,â 49.
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110)
âPeter B. Brown, âThe Military Chancellery: Aspects of Control During the Thirteen Yearsâ War,â Russian History 29, no. 1 (Spring 2002): 22-23.
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111)
âPeter B. Brown, âCommand and Control in the Seventeenth Century Russian Army,â in Warfare in Eastern Europe, 1500-1800, ed., Brian J. Davies (Brill: Leiden, Boston, 2012), 291.
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116)
âBrown, âHow Muscovy Governed,â 523; Peter B. Brown, âMuscovite Government Bureaus,â Russian History 10, no. 3 (1983): 314-15.
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117)
âSee S.N. Eisenstadt, The Political Systems of Empires. The Rise and Fall of the Historical Bureaucratic Societies (New York: the Free Press, 1969), 277-78.
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119)
âRichard Hellie ed., trans., âMuscovite-Western Commerical Relations,â Readings for Introduction to Russian Civilization. Muscovite Society (Chicago: Syllabus Division, the College, the University of Chicago, 1967), 63, 66-73, 79, 81-87. J.T. Kotilaine, who like Hellie notes certain retrograde, behavioral characteristics of the Russian merchantry, shows that the Muscovite merchantryâalbeit at glacial paceâdemonstrated modest adaptability in competition with foreign merchants as the seventeenth century wore on (J.T. Kotilaine, Russiaâs Foreign Trade and Economic Expansion in the Seventeenth Century. Windows on the World [Leiden, Boston: Brill, 2005], 215-29). Erika Monahan Downs, in her introductory commentary and absorbing historiographical review (âDestabilizing the Early Modern Failure Narrative: a Historiographical Critiqueâ), unequivocally argues for treating the first-rank merchants or gosti and the second-rank merchants or gostinaia sotnia (the Merchant Hundred) on their own terms and for rigorously eschewing exogenous, negative tropes characterizing Muscovyâs merchants (Downs, âTrade and Empire,â 17-38, 75-103).
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120)
âSee Peter B. Brown, âNeither Fish nor Fowl: Administrative Legality in Mid- and Late-Seventeenth-Century Russia,â Jahrbücher für Geschichte Osteuropas 50, no. 1 (2002): 1-21; id., âGuarding the Gate-Keepers: Punishing Errant Rank-and-File Officials in Seventeenth-Century Russia,â Jahrbücher für Geschichte Osteuropas 50, no. 3: 224-45.
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125)
âJames Cracraft, The Revolution of Peter the Great (Cambridge, Massachusetts; London: Harvard University Press, 2003), 101. Ryan notes that Arifmetikaâs pages also included the old alphanumerals (Ryan, âScientific Instruments in Russia,â 373).
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126)
âJeremy Black, European Warfare, 1494-1660 (London, New York: Routledge, 2002), 51-52; Elżbieta Jung, Robert PodkoÅski, âRichard Kilvington on Proportions,â in Mathématiques et théorie du movement (XIVe-XVIe siècles), eds., Joël Biard, Sabine Rommevaux (Villeneuve dâAscq: Presses Universitaires du Septentrion, 2008), 81-101; Edith Dudley Sylla, âCalculationes de motu locali in Richard Swineshead and Alvarus Thomas,â in Matématiques et théorie du movement, 131-46; Jean-Jacques Brioist, Pascal Brioist, âHarriot, lecteur dâAlvarus Thomas et de Niccolo Tartaglia,â in Mathématiques et théorie du movement, 147-71; Evelyne Barbin, âLa recherche galiléenne de la trajectoire des projectiles,â La révolution mathématique du XVIIesiècle (Paris: Ellipses, 2006), 84-101; id., âLa méthode des cercles tangents de Descartes,â in La révolution mathématique, 166-73. Kilvington deserves a monument for âHis most intriguing and successful approach to physical problems was to utilize the science of ratios in the context of the problem of motion.â (Jung and PodkoÅski, âKilvington on Proportions,â 81).
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127)
âP.L. Griffiths, âHow Napier Discovered Logarithms,â Mathematical Discoveries 1600-1750 (Ilfracombe: Arthur H. Stockwell, 1977), 7-12.
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128)
âAround 1980, when the Sandanista guerillas from Nicaragua began penetrating into Honduras, Honduran governmental officials and their American military advisers expressed astonishment over the innumeracy of Honduran recruits and even of Honduran officer candidates over their inability to calibrate rifle and artillery gunsights. The Honduran government then tried to counter this deficiency through a rushed program of remedial education for its military inductees. This regimeâs response to dire military exigency brings to mind the Petrine military education reforms to increase the level of minimal literacy and numeracy to its power elite, the dvorianstvo (the former Muscovite upper and middle service classes), who were military officers, landowners, serf-holders, and administrative officials.
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129)
âT. Rainov, Nauka v Rossii xi-xvii vekov. Ocherki po istorii do nauchnykh i estestvenno-nauchnykh vozzrenii na prirodu. Chasti i-iii (Moscow, Leningrad: AN SSSR, 1940), 278-88.
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âMax J. Okenfuss, The Rise and Fall of Latin Humanism in Early-Modern Russia. Pagan Authors, Ukrainians, and the Resiliency of Muscovy (Leiden, Boston, Cologne: E.J. Brill, 1995), 4-6, 14-17, 21-44. The full text of Peterâs December 19, 1699 edict on dating years from the birth of Christ is: âAs of 1 January 1700, years in all documents shall be written from the birth of Christ and not from the creation of the world. Years shall be written and counted from 1 January of the seven thousand two hundred and eighth year (7208âi.e., 1700), and the year 1700 shall be calculated from the birth of Our Lord God and Savior, Jesus Christ, in verdict resolutions and governmental messages and full-length documents and in all our Great Sovereignâs edicts on sundry matters (written up) in the Military Chancellery and in all other chancelleries; in any official documents (written up) on town-squares; and in copies of documents and resolutions and annual itemization and verdict resolution lists and in any governmental and communal paperwork from governors (stationed) in towns. And one year later, as of 1 January in the coming seven thousand two hundred and ninth year (1701), the dating from 1 January of the one thousand seven hundred and first year (1701) and for the following years shall be done from the birth of Christ in the same manner. And beginning from the new year in the month of January (1700), the months and dates that follow shall without fail be written chronologically up to (every) January for the following years, calculating years from the birth of Christ in the same manner. And we, the Great Sovereign, have ordered this done because many neighboring Christian peoples, who harmoniously uphold with us the Orthodox, Christian, Eastern Faith, write years by dating from the birth of Christ. And if whoever will wish to write (dates) from the creation of the world, they shall write both years, as they wish, from the creation of the world and from the birth of Christ, one after the otherâ (PSZ, 3, no. 1735: 680-81). This translation is the authorâs.
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Muscovite Arithmetic in Seventeenth-Century Russian Civilization: Is It Not Time to Discard the âBackwardnessâ Label?*
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Muscovite civilization utilized Byzantine-Greek alphanumerals for its mathematical symbols. Occasionally derided by historians for being retrograde in comparison to the Hindu-Arabic numerals sixteenth- and seventeenth-century Europe adopted, Muscovyâs alphanumerals were versatile and suitably contoured to perform a variety of computational tasks. Muscovite alphanumerals were an integral part of early Moderen Russiaâs administrative culture, and played a prominent role in fostering the experiential knowledge underlying the educational achievements of the Imperial Period. Though they lacked the zero and the decimal, Muscovites still had a reasonable grasp of the base-ten system, and comprehended well basic arithmetical skills and relationship properties, less so equational ones. The Russians developed complex abaci well suited for commercial transactions, large-scale construction, military inventories and payrolls, and the land registry, to name a few. These instruments manipulated an extensive variety of weights, measures, linear distances, area dimensions, volume measurements, and currency. Muscovite arithmetic was a prominent factor assisting in the advancement of critical thinking skills in 1600âs Russia. Nonetheless, as the seventeenth century wore on, sociological, educational or pedagogical, military scientific, administrative, and cultural arguments or interactive phenomena came to bear and increasingly found the Muscovite algorithmic symbols wanting. In 1699 the government decreed that Hindu-Arabic numerals henceforth were to be used in official documents throughout the country. Directly and indirectly, the complex thought processes bound up when operating with Muscovite alphanumerals were one impetus for the further unfolding of Russian civilization after 1700.
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