Abstract
No definitive history of the Russian abacus schyoty (cчёты) has ever been written; the hypothesis about its Asian origin stated by a number of Western authors was strongly opposed by I. G. Spasskiĭ (1904–1990) in his book-length article of 1952. The chapter is devoted to two topics: (1) the extant sources relevant to the history of the Russian instrument and (2) the operations with common fractions performed with it as described in Russian arithmetical manuals of the seventeenth century. The author concludes that the theory suggesting that the instrument was imported to Russia from the Golden Horde is worth to be explored, and offers an interpretation of the descriptions of operations with common fractions performed with the instrument.
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Notes
- 1.
This name of the instrument is relatively recent. Its depiction in the mathematical manuscripts of the seventeenth century are accompanied with the caption “дщица щетная (= счетная”) [dshchitsa shchetnaya], that is, “counting board”; see Spasskiĭ (1952, pp. 322–327, Figs. 15–20). In sources dated of the seventeenth century the instrument was also referred to as “счет” [schyot] or “дощаный счет” [doschchanyĭ schyot], that is, “counting” or “board counting,” and this word was used in singular even when the instrument most probably contained two counting surfaces joined together; see Spasskiĭ (1952, pp. 314–320). Spasskiĭ also reports that the term “доска” [doska] (“board”) applied to the instrument was still in use as late as the first half of the nineteenth century (1952, p. 318); by that time the term “schyoty” (i.e., literally “counts” or “reckonings,” in plural) was already widely used; see, for example, Orlitskiĭ (1830a, b) and Tikhomirov (1830).
- 2.
The number of bars in the upper part of the instrument varied considerably. For example, the schyoty made in 1830s or 1840s shown in (Spasskiĭ 1952, p. 392, Fig. 32) had only six bars in the upper part, while the lower part was the same as that of the instrument depicted in Kiryushin (1925) and reproduced in Fig. 1, i.e., had four bars with 4, 10, 10, and 4 beads, counting from the bottom. There existed modifications of the schyoty with 20 bars and 10 beads on each bar (Spasskiĭ 1952, p. 396, Fig. 33), 11 bars with 10 beads on each bar (ibid., p. 400, Fig. 36), and so on.
- 3.
The structure of the Russian instrument is different from that of the Chinese one: the numbers of beads on the bars are not the same, and the Russian instrument never had two sections used for beads of values 1 and 5, as did the Chinese one (even though the early specimens of the Russian instrument did have a vertical bar separating beads into two groups having different values; see below). Moreover, the positions in which the instruments were set on a flat surface for use were different, the Russian one was placed “vertically” (that is, the operator faced the short side of the frame) while the Chinese one was always set “horizontally.” The structure of the instrument and its orientation relative to the operator thus determined different procedures for performance of arithmetical operations.
- 4.
Spasskiĭ (1952) advanced several arguments against this hypothesis; he claimed that the instrument had no connection with the Chinese suanpan and was invented in Russia independently. Unfortunately, his work remained virtually unknown outside USSR/Russia; for rare exceptions, see Ryan (1972), Ryan (1991, esp. see pp. 373–374), and Burnett and Ryan (1998). On Spasskiĭ and his study of the schyoty see below.
- 5.
- 6.
- 7.
Spasskiĭ (1952, p. 275, esp. see footnote 2).
- 8.
- 9.
On the book and its author see the introductory note of the Russian translator (A. Stankevich) in Reutenfels (1905, pp. I–X). The dates of life of the author are unknown . Stankevich conjectures that Jacob Reutenfels stayed in Russia in 1670–1672 and suggests that the original text of Reutenfels may have been written in an unidentified language and translated into Latin by a German scholar in the late 1670s; the Latin translation was published in Padova in 1680, while the original text was lost.
- 10.
It is not clear who are the “Ta(r)tars” mentioned here. Apparently, the author distinguishes them from the “Chinese”, yet the term still may have been used to refer to Manchus who at that time governed China. There is, however, another possibility, namely, that the author referred to the Tatars inhabiting the territory of the present-day Russia, esp. the basin of lower Volga and areas to the East of it (including Ural mountains and Siberia).
- 11.
“Считают они посредством камешков и кораллов, нанизанных на проволоку и расположенных в два ряда, на подобие Tатар и Kитайцев.” (Reutenfels 1906, pp. 158–159).
- 12.
Spasskiĭ (1952, p. 411) made the following comment on Reutenfels’ description: “Reutenfels only briefly remarked that [the Russian shyoty] are designed ‘similarly to the Chinese or Tatar [ones]’”. It is possible that Spasskiĭ did not have access to the original text of the Reutenfels and used the translation of A. Stankevich (Reutenfels 1905–1906) instead. It is also interesting that in his quote Spasskiĭ (most probably, intentionally) changed the order and the connective found in the original (“Chinese or Ta(r)tar” instead of “Ta(r)tar and Chinese”).
- 13.
On Witsen’s life and work, see the monograph of Peters (2010); see also Peters (1994) and Hoving (2012). On Witsen’s travels to Russia, his personal connections there (among his Russian acquaintances was, for instance, Peter the Great) and his work see the introductory article of the recent three-volume annotated Russian translation of Witsen’s work (2010).
- 14.
I would like to express my gratitude to Professor Jan van Maanen who kindly helped me obtain access to the 1692 edition of Witsen’s book preserved in the Library of Utrecht University.
- 15.
Witsen (1692, p. 472).
- 16.
Золотая Орда [Zolotaya Orda] in Russian, hence “Solitaja Orda” of Witsen.
- 17.
The English translation is mine; it is based on a Russian translation published in Witsen 2010. Its last two sentences read as follows: “[…] Этот старый Строганов привез в Россию, как говорят, счеты, или арифметику, которые они еще употребляют до сего дня. Это костяные бусинки, нанизанные на железные прутики.” The modern translator thus interpreted the Dutch word cyfer (i.e., literally, digit, figure, numeral) as “schyoty”.
- 18.
- 19.
The majority of researchers agree that the “Old Saraĭ” was located near the present-day Selitrennoe Gorodishche (Селитренное городище) in Astrakhan region, while the location of the “New Saraĭ” remains a matter of controversy. Some authors suggest that it was located near the present-day Russian city of Volgograd (better known in Western Europe as Stalingrad), while others argue against this theory; for more details see (Egorov 1985; Il’ina 2005; Pachkalov 2009a, b; Zaĭtsev 2006).
- 20.
Gerhard Friedrich Müller, 1705–1783; his name was traditionally transliterated in Russian as “Герард Фредерик Миллер” (hence “Miller”).
- 21.
Spasskiĭ (1952, p. 416) mentions a genealogical tree of Stronganovs found in a document of the eighteenth century; according to it, Spiridon was born in 1362. In turn, the genealogical tree reproduced in (Ustryalov 1842) suggests that the first Stroganov passed away in 1395. These dates of life of Spiridon are consistent with the claim stating that he lived during the reign of Dmitriĭ Donskoĭ (1359–1389).
- 22.
The legend of a nobleman escaping from the Golden Horde in the mid-fourteenth century to be baptized as Eastern Orthodox Christian can be to some extent corroborated by the fact that Islam became the state religion of the Golden Horde in 1320s, during the rule of Uzbek-Khan (r. 1313–1341), and therefore in the mid-fourteenth century some Tartar Christians indeed may have had strong reasons to move to the Grand Duchy of Moscow.
- 23.
Scherer (1792, v. 1, p. 153).
- 24.
Scherer (1792, v. 1, pp. 153–154, note 1).
- 25.
For example, Kiryushin (1925) in the very beginning of his introductory chapter writes “It is not known when exactly did the Russian schyoty appear. There is a hypothesis that the idea of the design [of the instrument] was borrowed from the Chinese counting device Suan-pan in the 16th century by the merchants Stroganovs who lived at the border of Siberia” (p. 9). Interestingly enough, this author does not notice that the lands possessed by the Stroganovs were located near the Western, and not Eastern, border of Siberia, that is, quite far from China.
- 26.
“Китайский суан-пан, возникновение которого относится к глубокой древности …” (Chinese suanpan, whose origin is dated of the high antiquity…), see Spasskiĭ (1952, p. 291). Apparently, Spasskiĭ was not aware that the time of the creation of the Chinese suanpan is a matter of controversy, and the earliest evidence of its use of China is a depiction of an object visually resembling the modern suanpan and found in a painting produced in the 12th century. A description of a counting instrument similar to the Roman abacus is found in the Chinese mathematical treatise Shu shu ji yi 數術記遺 (Procedures of “numbering” recorded to be preserved [for posterity]), compiled by Xu Yue 徐岳 (b. ca. 185 – d. ca. 227 AD), with commentaries of Zhen Luan 甄鸞 (b. ca. 500 – d. after 573 AD) (Xu 1993); this device is discussed below. However, dating the treatise of Xu Yue is a quite complex matter; for more details, see, for example, Volkov (1994).
- 27.
On this school see Freiman and Volkov (2015) and literature cited there.
- 28.
Spasskiĭ 1952, p. 368, n. 3.
- 29.
In his paper of 1972, William F. Ryan stated that “The evidence… points to a strong probability [that the instrument was brought from Russia in 1618 by John Tradescant the Elder]… [but one] cannot definitely exclude possible alternative sources [of the instrument]” (pp. 85–86); for instance, it is possible that the instrument was purchased in 1650s from an individual working for Muscovy Company (p. 86).
- 30.
Ryan (1972, pp. 83, 85).
- 31.
This system of representation of numbers is described not only in the appendix of 1744, but in the book itself (p. 119).
- 32.
Interesitngly enough, Snelling (1769, p. 16) also reports that the units occupy the uppermost position, the dozens are set below them, etc. Moreover, he believed that only the beads pushed to the right side are counted. He suggests that this disposition can be modified: the units can be set below, the dozens above them, etc. (yet still only rightmost beads are counted); the picture at the end of his paper shows the latter disposition. Each bar has nine beads.
- 33.
See, for example, the Chinese mathematical treatise Pan zhu suan fa 盤珠算法 (Computational methods for abacus [lit. “pearls in tray”]) of 1573 (Xu 1993 [1573], juan 1, p. 20b).
- 34.
These terms are transcribed by von Haven as Hao, Li, Füen, Tsïen, and Learg, respectively.
- 35.
“В приказах были […] сливяные и вишневые косточки, при помощи которых производился счет” (In the offices there were seeds of plum and cherries used for computations) (Shtaden [Staden] 1925, p. 83); “Счет ведут при помощи сливяных косточек” ([They] conduct counting with the help of plum seeds) (Shtaden [Staden] 1925, p. 123).
- 36.
- 37.
It should be noted, however, that the diagram of this kind is found in only one picture of this richly illustrated chronicle; moreover, Simonov himself failed to provide an interpretation of the configurations of dots shown in the diagram (in his monograph of 1977 he suggests that these configuration “may have represented results of a divination,” p. 63).
- 38.
Schärlig (2001) mentions two representations of the instrument of this type found in Greece, the first of them is found in a depiction dated of ca. 280 BC (pp. 88–89), and the second one is drawn on a piece of marble dated of the 1st century AD (pp. 90–91).
- 39.
On the archaic Russian abacus see also Vilenchik (1984).
- 40.
Guo and Liu (2001, p. 450).
- 41.
The problem is rather complex given that the dates of compilation of the Chinese treatise (the 2nd century) and of the commentary in which a detailed description of the instrument is provided (the 6th century) are not completely certain, since the original treatise was lost by the late first millennium AD and the version available nowadays is based on a manuscript found in the late 12th century. For more details concerning this treatise see, for example, Volkov (1994).
- 42.
For a bibliography of the extant Russian mathematical manuscripts see Shvetsov (1955).
- 43.
On “sokha script” see Shvetsov (1966).
- 44.
Spasskiĭ (1952, p. 338, Fig. 21).
- 45.
In this table “[A]” means that the term A should be added, and “<B>” means that the term B should be removed or replaced.
- 46.
- 47.
Tsaĭger suggests that in the Table shown in Fig. 12 the scribe forgot to mention that in the first line the “price” of one seed in the first column equals to 1/3 (Tsaĭger 2010, p. 54). However, the “price” of one seed mentioned in the fifth column of the same line equals to 1/4, and most likely we are dealing here not with an omission of a scribe but with a particular convention used only in the cases of fractions with denominators 3 and 4 and numerators larger than 1.
- 48.
This list of seven identities was for the first time reconstructed by Tsaiger (2010, p. 56, Fig. 24).
- 49.
- 50.
- 51.
Picture for this identity is not found in Anon. (1865).
- 52.
The picture of the schyoty provided in Anon. 1865 for this identity does not contain a bead in the position of 1/24; the picture reproduced in Spasskiĭ (1952, p. 344, Fig. 25) correctly shows a bead in this position.
- 53.
The manuscript Anon. (1865) erroneously misses a bead in the position of 1/6.
- 54.
The manuscript Anon. (1865) erroneously misses a bead in the position of 1/8.
- 55.
The manuscript Anon. (1865) erroneously misses a bead in the position of 1/8.
- 56.
By “shiftning up” I mean a transformation of a given configuration consisting in placing active beads in n-th wire (counted from the top) on (n − 1)-th wire; “shifting down” is moving active beads from n-th wire to (n + 1)-th wire.
- 57.
One can only regret that the traditional Russian schyoty with sections for fractions disappeared by the early 18th century and thus remained unknown to the Western educators of the 19th century who argued for the use of counting devices (and, in particular, for the use of the Russian schyoty) in school classroom. It would be interesting to see what results might have been obtained if teaching common fractions were performed with the help of the original instrument.
References
A. Russian Mathematical Textbooks of the 17th Century
[Anon.] 1629 [1853]. Книгa coшнoгo пиcьмa 7137 гoдa [Kniga soshnogo pis’ma 7137 goda] (Book of sokha script of the year 7137 [=1629]). Transcribed by G. Trekhletov. Bpeмeнник Импepaтopcкaгo Mocкoвcкaгo oбщecтвa иcтopии и дpeвнocтeй poccийcкиx [Vremennik Imperatorskago Moskovskago obshchestva istorii i drevnostei rossiĭskikh], Book 17, pp. 33–65.
[Anon.] 1865. Книгa coшнoмy и вытнoмy пиcьмy [Kniga soshnomu i vytnomu pis’mu] (Book of sokha and vyt’ script). A manuscript acquired by the library of the Troitsk-Sergiev Monastery in 1865.
[Anon.] 1879. Cчётнaя мyдpocть [Schyotnaya mudrost’] (Counting Wisdom). Sankt-Peterburg: Obshchestvo lyubitelei drevnei pis’mennosti.
B. Primary Works in Other Languages
Du Halde, Jean-Baptiste. 1735. Description Geographique, Historique, Chronologique et Physique de l’Empire de la Chine et de la Tartarie Chinoise. Paris: Le Mercier.
Guo Shuchun 郭書春 (ed.) 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙 (Comprehensive compendium of classical texts related to science and technology in China). Shuxue juan 數學卷 (Mathematical section). Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
Guo Shuchun 郭書春, and Liu Dun 劉鈍 (eds.) 2001. Suanjing shi shu 算經十書 (Ten mathematical treatises). Taibei: Jiuzhang Publishers.
von Haven, Peder von. 1743. Reise udi Rusland. Kjøbenhavn: Christoph Georg Glasing.
von Haven, Peter von. 1744. Reise in Rußland [German translation of von Haven (1743)]. Coppenhagen: Gabriel Christian Rothe.
Korb, Joanne Gergio [=Johann Georg]. 1700. Diarium itineris in Moscoviam. Viennae: Leopoldi Voigt.
Miller, Gerard Frederik [Mиллep, Гepapд Фpeдepик] [=Müller, Gerhard Friedrich]. 1750. Opisanie Sibirskago Tsarstva i vseh proizoshedshih v nem del ot nachala, a osoblivo ot pokoreniya ego Rossiĭskoi Derzhave, po sii vremena [Oпиcaниe Cибиpcкaгo Цapcтвa и вcex пpoизoшeдшиx в нeм дeл oт нaчaлa, a ocoбливo oт пoкopeния eгo Poccийcкoй Дepжaвe, пo cии вpeмeнa] (Description of the Siberian Kingdom and all the events that happened in it from the very beginning, and especially from the subjugation of it by the Russian State, until the present times). Sankt-Peterburg: Academy of Sciences.
Olearius, Adam. 1656. Relation du voyage de Moscovie, Tartarie, et Perse. Paris: Francois Clouzier.
Oleariĭ [Oлeapий] [=Olearius], Adam. 1906. Opisanie puteshestviya v Moskoviyu i cherez Moskoviyu v Persiyu i obratno [Oпиcaниe пyтeшecтвия в Mocкoвию и чepeз Mocкoвию в Пepcию и oбpaтнo] (Description of the travel to Muscovy and through Muscovy to Persia and back). Translated into Russian by A. M. Lovyagin. Sankt-Peterburg. [Reprint: Smolensk: Rusich, 2003].
Perry, John. 1716. The state of Russia under the present Czar. London: B. Tooke.
Perry, Jean [=John]. 1717. État présent de la Grande-Russie. La Haye: Henry Dusauzet.
Reutenfels, Iacobus. 1680. De Rebus Moschoviticis ad Serenissimum Magnum Hetruriae Ducem Cosmum Tertium. Patauii: Petri Mariae Frambotti (The name of the author is not found on the front page; it is mentioned in the Admonitio section in the beginning of the book.).
[Reutenfels, Iacobus] Yakov Rejtenfel’s [Якoв Peйтeнфeльc]. 1905–1906. Skazaniya svetleĭshemu gertsogu Toskanskomu Ko’zme tret’emu o Moskovii (Paduya, 1680 g.) [Cкaзaния cвeтлeйшeмy гepцoгy Tocкaнcкoмy Кoзьмe тpeтьeмy o Mocкoвии (Пaдyя, 1680 г.)] (Accounts [presented] to the illustrious Duke of Tuscany, Cosimo the Third, about the Muscovy (Padua, anno 1680)). [Russian translation of Reutenfels 1680 by Aleksei I. Stankevich.] Чтeния в Импepaтopcкoм oбщecтвe Иcтopии и Дpeвнocтeй Poccийcкиx пpи Mocкoвcкoм Унивepcитeтe. Mocквa: Tипoгpaфия oб-вa pacпpocтpaнeния пoлeзныx книг. [Part 1:] 1905, Book 3, Sect. II, pp. I–X, 1–128; [Part 2:] 1906, Book 3, Sect. III, pp. 129–228.
Scherer, Jean-Benoît [Johann Benedikt]. 1792. Anecdotes intéressantes et secrètes de la Cour de Russie, tirées de ses archives. Londre et Paris: Buisson.
Snelling, Thomas. 1769. A view of the origin, nature, and use of jettons or counters. London.
Shtaden, Genrikh [Штaдeн, Гeнpиx = Heinrich von Staden]. 1925. O Mocквe Ивaнa Гpoзнoгo: Зaпиcки нeмцa oпpичникa [O Moskve Ivana Groznogo: Zapiski nemtsa oprichnika] (About Moscow of Ivan the Terrible: Notes of a German oprichnik). Moscow: Sabashnikovs Publishers [an annotated Russian translation of a manuscript copy of the 16th century preserved in a German archive].
Witsen, Nicolaes. 1692. Noord en Oost Tartarye, Ofte Bondigh Ontwerp Van Eenige dier landen, en volken, zo als voormaels bekent zyn geweest. […] Zedert nauwkeurigh onderzoek van veele jaren, en eigen ondervindinge beschreven, getekent, en in ‘t licht gegeven, door Nicolaes Witsen. Amsterdam. [Second edition (under slightly different title): Amsterdam: M. Schalekamp, 1705; reprint 1785].
Witsen, Nicolaas (Bитceн, Hикoлaac). 2010. Ceвepнaя и вocтoчнaя Tapтapия: включaющaя oблacти, pacпoлoжeнныe в ceвepнoй и вocтoчнoй чacтяx Eвpoпы и Aзии [Severna͡ia i vostochna͡ia Tartar͡iia: vk͡liucha͡iushcha͡ia oblasti, raspolozhennye v severnoĭ i vostochnoĭ chas͡tiakh Evropy i Azii]. Amsterdam: Pegasus Publishers. [An annotated Russian translation of Witsen 1785].
Xu, Xinlu 徐心魯. 1993 [1573]. Pan zhu suan fa 盤珠算法 (Computational methods for abacus [lit. “pearls in tray”]). In Guo (1993), Vol. 2, pp. 1143–1164.
Xu, Yue 徐岳. 1993. Shu shu ji yi 數術記遺 (Procedures of “numbering” recorded to be preserved [for posterity]). In Guo (1993), Vol. 1, pp. 347–351; Guo and Liu 2001, pp. 445–452.
C. Russian and Soviet School Manuals Related to Schyoty
Kiryushin, E. D. [Киpюшин, Eфим Дaнилoвич]. 1925. Bычиcлeния нa cчётax [Vychisleniya na schyotakh] (Calculations with the schyoty), 6th ed. Moscow: Kooperativnoe izdatel’stvo.
Orlitskiĭ, D. [Opлицкий, Д.] 1830a. Умнoжeниe и дeлeниe чиceл нa cчeтax, c пpиcoвoкyплeниeм ocoбыx пpaвил cлoжeния и вычитaния [Umnozhenie i delenie chisel na schyotakh, s prisovokupleniem osobykh pravil slozheniya i vychitaniya] (Multiplication and division of numbers on the schyoty, with additional special methods of addition and subtraction). Caнкт-Пeтepбypг: Гpeч.
Orlitskiĭ, D. [Opлицкий, Д.] 1830b. Умнoжeниe и дeлeниe чиceл нa cчётax [Umnozhenie i delenie chisel na schyotakh] (Multiplication and division of numbers on the schyoty). Syn Otechestva [Cын Oтeчecтвa], 131: 297–307.
Tikhomirov, Petr [Tиxoмиpoв, Пeтp Bacильeвич]. 1830. Apифмeтикa нa cчётax или лeгчaйший cпocoб пpoизвoдить вce apифмeтичecкиe дeйcтвия нa cчётax, ycoвepшeнcтвoвaнныx гeнepaл-мaйopoм г. Cвoбoдcким [Arifmetika na schyotah ili legchaishiĭ sposob proizvodit’ vse arifmeticheskie deĭstviya na schyotakh, usovershenstvovannykh general-maĭorom g. Svobodskim] (Arithmetic on the schyoty, or the easiest way to produce all the arithmetical operations of the schyoty improved by general-major Mr Svobodskoi). Caнкт-Пeтepбypг: Mopcкaя типoгpaфия.
D. Secondary Works
Bobynin, Viktor [Бoбынин, Bиктop Bиктopoвич]. 1884. Cocтoяниe мaтeмaтичecкиx знaний в Poccии дo XVI вeкa [Sostoyanie matematicheskikh znaniĭ v Rossii do XVI veka] (The state of mathematical knowledge in Russia prior to the 16th century; in Russian). Жypнaл Mиниcтepcтвa Hapoднaгo Пpocвeщeния [Zhurnal Ministerstva Narodnago Prosveshcheniya]. Journal of the Ministry of Education, 132(2): 183–209.
Bobynin, Viktor [Бoбынин, Bиктop Bиктopoвич]. 1886. Oчepки иcтopия paзвития физикo-мaтeмaтичecкиx знaний в Poccии. XVII cтoлeтиe [Ocherki istoriya razvitiya fiziko-matematicheskikh znaniĭ v Rossii. XVII stoletie] (Outlines of the development of physics and mathematics in Russia. The 17th century; in Russian). Moscow: Mamontov.
Burnett, Charles, and William F. Ryan. 1998. Abacus (Western). In Instruments of science: An historical encyclopedia, ed. Robert Bud, Deborah J. Warner, 5–7. London and New York: Garland.
Egorov, Vadim L. [Eгopoв, Baдим Лeoнидoвич]. 1985. Иcтopичecкaя гeoгpaфия Зoлoтoй Opды в XIII-XIV в.в. [Istoricheskaya geografiya Zolotoĭ Ordy v XIII-XIV vv.] (Historical geography of the Golden Horde in the 13th–14th century; in Russian). Moscow: Nauka.
Freiman, Viktor, and Alexei Volkov. 2012. Common fractions in L. F. Magnitskiĭ’s Arithmetic (1703): Interplay of tradition and didactical innovation. Paper delivered at the 12th International Congress on Mathematics Education, Seoul, South Korea, July 8–15, 2012.
Freiman, Viktor, and Alexei Volkov. 2015. Didactical innovations of L. F. Magnitskiĭ: setting up a research agenda. International Journal for the History of Mathematics Education, 10(1): 1–23.
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Volkov, A. (2018). Counting Devices in Russia. In: Volkov, A., Freiman, V. (eds) Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Mathematics Education in the Digital Era, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-73396-8_12
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