Muscovite Arithmetic in Seventeenth-Century Russian Civilization: Is It Not Time to Discard the “Backwardness” Label?

In: Russian History

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, Muscovy’s alphanumerals were versatile and suitably contoured to perform a variety of computational tasks. Muscovite alphanumerals were an integral part of early Moderen Russia’s 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 1600’s 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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  • 2)

    Florovsky“The Problem of Old Russian Culture” 133-36.

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     See Robert RomanchukByzantine Hermeneutics and Pedagogy in the Russian North: Monks and Masters at the Kirillo-Belozerskii Monastery 1397-1501 (Toronto: University of Toronto Press2007).

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     E.g. Peter B. Brown“Peering into a Muscovite Turf-War (How Do We Even Know It’s There?): Boyar Miloslavskii and the Auditing Chancellery,” Russian History 25 nos. 1-2 (Spring-Summer 1998): 141-53; id. “Salaries and Economic Survival: the Service Land Chancellery Clerks of Seventeenth-Century Russia” Jahrbücher für Geschichte Osteuropas 51 (2003): 32-67; id. “Bureaucratic Administration in Seventeenth-Century Russia” in Modernization of Muscovy eds. Jarmo Kotilaine Marshall Poe (London and New York: RoutledgeCurzon 2004) 57-78; id. “How Muscovy Governed: Seventeenth-Century Russian Central Administration” 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 “The Military Revolution Administrative Development and Cultural Change in Early Modern Russia” Journal of Early Modern History 2 no. 3 (August 1998) 247-73; id. “Muscovite Personnel Records 1475-1550: New Light on the Early Evolution of Russian Bureaucracy” Jahrbücher für Geschichte Osteuropas 45 no. 3 (1997): 361-77; id. “Elite Service Registry in Muscovy 1500-1700” Russian History 21 no. 3 (Fall 1994): 253-83.

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    Nigel WhealeWriting and Society. Literacy Print and Politics in Britain 1590-1660 (London, New York: Routledge1999) 2. These are aggregate figures inclusive of both genders and all social strata (ibid.).

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  • 24)

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  • 28)

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  • 30)

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  • 31)

     See L.V. CherepninRusskaia paleografiia (Moscow: Gosizdatpolitlit1956) 373.

  • 34)

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  • 35)

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  • 36)

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  • 38)

    SimonovMatematicheskaia i kalendarno-astronomicheskaia mysl’59-60. One of the reviewers noted that “Both 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).”

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  • 39)

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  • 41)

    Lakoff NuñezWhere Mathematics Comes From19 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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  • 44)

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  • 46)

    Peter J. BentleyThe 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)

    W.F. Ryan“Scientific Instruments in Russia from the Middle Ages to Peter the Great,” Annals of Science. The History of Science and Technology from the Thirteenth Century48 no. 4 (July 1991) 373.

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  • 49)

    Helmuth Gericke“Die Zeit von 1500 bis 1637,” Mathematik im Abendland. Von den römischen Feldmessern bis zu Descartes (Berlin Heidelberg New York: Springer-Verlag 1980) 225-96.

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  • 51)

    V.A. Petrova comp.Paleograficheskii al’bom. Uchebnyi sbornik snimkov s rukopisei russkikh dokumentov xiii-xviii vv. (Leningrad: Izdatel’stvo LGU1968) 19 33 41 49; M.N. Tikhomirov A.V. Murav’ev Russkaia paleografiia (Moscow: Vysshaia shkola 1966) 4-5; L.V. Cherepnin Russkaia khronologiia (Moscow: Glavnoe arkhivnoe upravlenie NKVD SSSR 1944) 20-24; Grigore Nandriş Handbook of Old Church Slavonic 2 pts. corrected ed. (London: Athlone Press 1965) 1: 3; Simonov Matematicheskaia i kalendarno-astronomicheskaia mysl’ 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)

    John Allen PaulosInnumeracy. Mathematical Illiteracy and Its Consequences (New York: Hill and Wang2001) 11 28 44 108-10 141-42; Arthur Benjamin and Michael Shermer Secrets of Mental Math. The Mathemagicians’s Guide to Lightning Calculation and Amazing Math Tricks (New York: Three Rivers Press 2006) 203-04.

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  • 56)

    W.F. Ryan“John Tradescant’s Russian Abacus,” Oxford Slavonic Papersn.s. 5 (1972): 87; id. “Scientific Instruments in Russia” 373. For an introduction to that appliance see Jesse Dilson The Abacus. The World’s 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. Martin’s Griffin 1968) 33-55 and Boyer Merzbach A History of Mathematics 225-27.

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  • 57)

    Ryan“John Tradescant’s Russian Abacus” 86.

  • 59)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 332 336.

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    GullbergMathematics: from the Birth of Numbers102-05.

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    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 336-37.

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    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 315-17. Variants are depicted in ibid. 306 308. “The origins of the doshchanoi or doshchatoi schet remain obscure. Western European counting boards may have parented” (reviewer’s comment). One pound is .454 kg (Pocket Oxford Dictionary 456).

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  • 71)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 328 347.

  • 72)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 305-21 328-29; W.F. Ryan “Abacus (Western)” 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 4840 square yards or 4046.9 square meters (Pocket Oxford Dictionary 9).

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  • 73)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 308 311-16.

  • 74)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 308-09 312-15 335-36. Possibly there were yet other configurations for the seventeenth-century complex calculations’ board (ibid. 319).

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  • 75)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 318.

  • 77)

    IushkevichIstoriia matematiki v Rossii28; Ryan “Scientific Instruments in Russia” 37.

  • 78)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 321-27 331-33.

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    IushkevichIstoriia matematiki v Rossii do 1917 goda15-16 28.

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    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 322-27.

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    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 322-27.

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    IushkevichIstoriia matematiki v Rossii28.

  • 86)

    Boyer MerzbachHistory of Mathematics69; Spasskii “Proiskhozhdenie i istoriia russkikh schetov” 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 “comes naturally” 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)

     See Lakoff NuñezWhere Mathematics Comes From94-95. 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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  • 88)

    Brown“How Muscovy Governed” 498-99.

  • 89)

    HellieEconomy and Material Culture of Russia243.

  • 90)

    VeselovskiiSoshnoe pis’mo1: 44-47 87-88 93-94. Nonetheless the nagging problem of land measurement units’ 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)

    Spasskii“Proiskhozhdenie i istoriia russkikh schetov” 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 chet’ (“the inhabited chet’) and zhivushchaia vyt’ (“the inhabited vyt’) introduced experimentally in the 1630’s 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 bol’shaia sokha and its fractionable units. Only with the household tax’s universalization in the late 1670’s did the government in an attempt to drum up more revenue abandon the older and by now obsolete bol’shaia sokha. Using households as units had one advantage namely they could not be fractionalized unlike amounts of land (Dmitrieva “Metrologiia” 489-90; N.A. Rozhkov Russkaia istoriia v sravnitel’no-istoricheskom osveshchenii. Osnovy sotsial’noi dinamiki 2d ed. [Petrograd Moscow: Kniga early 1920’s] 4 pt. 2: 88; N.V. Ustiugov “Finansy” 418-19).

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  • 92)

    VeselovskiiSoshnoe pis’mo2: 393; N.V. Ustiugov “Finansy” in Ocherki istorii SSSR. Period feodalizma xvii v. eds. A.A. Novosel’skii N.V. Ustiugov (Moscow: AN SSSR 1955) 420; G.V. Alferova “Matematicheskie osnovy russkogo gradostroitel’stva xvi-xvii vv.” in Estestvennonauchnye znaniia v Drevnei Rusi ed. R.A. Simonov (Moscow: Nauka 1980) 131; I.V. Levochkin “Ob izmerenii sel’skokhoziaistvennykh ugodii v tsentral’noi Rossii xvi-xvii vv.” in Estestvennonauchnye znaniia v Drevnei Rusi 134-35; Dmitrieva “Metrologiia” 487-91; M.M. Krom Spetsial’nye 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 (“black” 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 “Metrologiia” 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 peasants’ 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)

    VeselovskiiSoshnoe pis’mo1: 58; Hellie Economy and Material Culture 648. For more such examples see Veselovskii Soshnoe pis’mo 1: 50 72; 2: 432.

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  • 97)

    K.S. NosovRusskie kreposti kontsa xv-xvii v. (St. Petersburg: Fakul’tet filologii i iskusstv Sankt-Peterburgskogo gosudarstvennogo universiteta2009) 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: Longman—Pearson Education Limited 2000); Carol B. Stevens Russia’s 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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  • 99)

    NosovRusskie kreposti38.

  • 100)

    KostochkinFedor Kon’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)

    OrlovskiiSmolenskaia stena37; G.L. Kurzov O kreposti staroi (Smolensk: Gody 2003) 8-10. The 320000 figure for both the barrels of lime and the piles is coincidental. One foot is 12” or 30.48 cm and one ton (American or short ton) is 2000 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)

    Peter B. Brown“Muscovite Government Bureaus,” 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 Kon’ 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)

     See Valerie A. KivelsonCartographies of Tsardom: the Land and its Meanings in Seventeenth-Century Muscovy (Ithaca, New York: Cornell University Press2006). To read about how the Muscovites understood geometry and comprehended town-wall building techniques see G.V. Alferova “Matematicheskie osnovy russkogo gradostroitel’stva xvi-xvii vv.” 109-33; id. “Organizatsiia stroitel’stva gorodov v russkom gosudarstve xvi-xvii vekakh” Voprosy istorii 1977 no. 7: 50-66; id. “Gosudarstvennaia sistema stroitel’stva gorodov i osvoenie novykh zemel’ v xvi-xvii vv. (na primere goroda Kozlova i ego uezda)” Arkhitekturnoe nasledstvo 1979 no. 27: 1-16; id. “K voprosu o stroitel’stve gorodov Moskovskim gosudarstvom v xvi-xvii vv.” Arkhitekturnoe nasledstvo 1980 no. 28; A.A. Piletskii “Sistema razmerov i ikh otnoshenii v drevnerusskoi arkhitekture” Estestvennonauchnye znaniia v Drevnei Rusi 63-109. In that vein see also D.V. Liseitsev “K voprosu o Gorodovom prikaze kontsa xvi-nachala xvii veka” in Rossiiskaia real’nost’ kontsa xvi-pervoi poloviny xix vv.: ekonomika obshchestvennyi stroi kul’tura. 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)

    Hellie“The Costs of Muscovite Military Defense and Expansion” 49.

  • 110)

    Peter B. Brown“The Military Chancellery: Aspects of Control During the Thirteen Years’ War,” Russian History 29 no. 1 (Spring 2002): 22-23.

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  • 111)

    Peter B. Brown“Command and Control in the Seventeenth Century Russian Army,” in Warfare in Eastern Europe 1500-1800ed. Brian J. Davies (Brill: Leiden, Boston2012) 291.

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  • 112)

    Livio“Why Math Works” 82-83.

  • 115)

    PaulosInnumeracy28 104-05.

  • 116)

    Brown“How Muscovy Governed” 523; Peter B. Brown “Muscovite Government Bureaus” Russian History 10 no. 3 (1983): 314-15.

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  • 117)

     See S.N. EisenstadtThe Political Systems of Empires. The Rise and Fall of the Historical Bureaucratic Societies (New York: the Free Press1969) 277-78.

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  • 119)

    Richard Hellie ed. trans.“Muscovite-Western Commerical Relations,” 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 merchantry—albeit at glacial pace—demonstrated modest adaptability in competition with foreign merchants as the seventeenth century wore on (J.T. Kotilaine Russia’s 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 (“Destabilizing the Early Modern Failure Narrative: a Historiographical Critique”) 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 Muscovy’s merchants (Downs “Trade and Empire” 17-38 75-103).

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  • 120)

     See Peter B. Brown“Neither Fish nor Fowl: Administrative Legality in Mid- and Late-Seventeenth-Century Russia,” Jahrbücher für Geschichte Osteuropas 50 no. 1 (2002): 1-21; id. “Guarding the Gate-Keepers: Punishing Errant Rank-and-File Officials in Seventeenth-Century Russia” Jahrbücher für Geschichte Osteuropas 50 no. 3: 224-45.

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  • 125)

    James CracraftThe Revolution of Peter the Great (Cambridge, Massachusetts; London: Harvard University Press2003) 101. Ryan notes that Arifmetika’s pages also included the old alphanumerals (Ryan “Scientific Instruments in Russia” 373).

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  • 126)

    Jeremy BlackEuropean Warfare 1494-1660 (London, New York: Routledge2002) 51-52; Elżbieta Jung Robert Podkoński “Richard Kilvington on Proportions” in Mathématiques et théorie du movement (XIVe-XVIe siècles) eds. Joël Biard Sabine Rommevaux (Villeneuve d’Ascq: Presses Universitaires du Septentrion 2008) 81-101; Edith Dudley Sylla “Calculationes de motu locali in Richard Swineshead and Alvarus Thomas” in Matématiques et théorie du movement 131-46; Jean-Jacques Brioist Pascal Brioist “Harriot lecteur d’Alvarus Thomas et de Niccolo Tartaglia” in Mathématiques et théorie du movement 147-71; Evelyne Barbin “La recherche galiléenne de la trajectoire des projectiles” La révolution mathématique du XVIIesiècle (Paris: Ellipses 2006) 84-101; id. “La méthode des cercles tangents de Descartes” in La révolution mathématique 166-73. Kilvington deserves a monument for “His most intriguing and successful approach to physical problems was to utilize the science of ratios in the context of the problem of motion.” (Jung and Podkoński “Kilvington on Proportions” 81).

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  • 127)

    P.L. Griffiths“How Napier Discovered Logarithms,” Mathematical Discoveries 1600-1750 (Ilfracombe: Arthur H. Stockwell 1977) 7-12.

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  • 128)

    Around 1980when 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 regime’s 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.

  • 129)

    T. RainovNauka v Rossii xi-xvii vekov. Ocherki po istorii do nauchnykh i estestvenno-nauchnykh vozzrenii na prirodu. Chasti i-iii (Moscow, Leningrad: AN SSSR1940) 278-88.

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  • 131)

    StevensRussia’s Wars of Emergence229-30.

  • 134)

    Max J. OkenfussThe Rise and Fall of Latin Humanism in Early-Modern Russia. Pagan Authors Ukrainians and the Resiliency of Muscovy (Leiden, Boston, Cologne: E.J. Brill1995) 4-6 14-17 21-44. The full text of Peter’s December 19 1699 edict on dating years from the birth of Christ is: “As 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 (7208—i.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 Sovereign’s 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 other” (PSZ 3 no. 1735: 680-81). This translation is the author’s.

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