Hugo of Lerchenfeld: Toledan numerals
On Gerbert’s abacus’ see Evans 1977a, 1977b, 1979. The later trajectory of Arabic numerals in the Latin world is discussed in Lemay 1982, Burnett 1997 (pp. 48–53 with figures 23–5), EHAS s.v. ‘The influence of Arabic mathematics in the medieval West’ (by Allard), Berggren 2002, Kunitzsch 2005, Burnett 2002b, Chrisomalis 2010b (chapter 6), Weddell 2015 and Burnett 2020; see also Katz 2016 (chapter 1 by Folkerts and Hughes). For the Hispanic contribution see Lemay 1977; on Greek examples Allard 1977 and Wilson 1981, and on transmission through Antioch Burnett 2000 (pp. 65–66), 2003.
On the practices of calculation with the numerals, see also Allard 1990 and Folkerts 1970 (chapters 9–12), 2001.
The specific case I discuss appears in Arrighi 1968 and Nothaft 2014 (cf. Herreman 2001). Further information on Hugo of Lerchenfeld and the Regensburg annals is from von Fichtenau 1937 (see esp. pp. 321–4) and Geschichtsquellen des deutschen Mittelalters (http://www.geschichtsquellen.de) s.vv. ‘Annales Ratisponenses’, ‘Wolfgerus monachus Prufeningensis’.
Material on the later Latin spread of the numerals is from Mercier 1987, Schärlig 2010, Crossley 2013 and Danna 2019. ‘Argus, the noble counter’ is from Chaucer, The Book of the Duchess, line 435. Myths about resistance to the numerals are addressed in Nothaft 2020.
The account keeper: Counting on paper
On Maes see Suchtelen et al. 2019; The Account Keeper is discussed at pp. 92–5 and also in Rathbone 1951. See also Watkins et al. 1984, pp. 218–19 (by Robinson), Robinson 1996 and Eikemeier 1984, and the description at https://www.slam.org/collection/objects/38142/.
The classic studies of ciphering books are Ellerton and Clements 2012, 2014. ‘Divide twelve pounds’ and the subsequent two quotations are from an American cyphering book of the 1770s described in the latter; see pp. 67, 73, 77. The genre is also studied in Denniss 2012, and in Wardhaugh 2012 (chapter 3).
Caroline Molesworth: Counting the weather
Numerical tabulation is surveyed in Campbell-Kelly et al. 2003. Important discussions of the new ways of counting and quantification in the period are Otis 2013, Porter 1995 and Poovey 1998, as well as for the American case Cohen 1982.
The major discussion of weather diaries is Golinski 2007. Specific examples are discussed in Golinski 2001, Brázdil et al. 2008, Lee et al. 2010, Lorrey 2012, Zhang 2013, Domínguez‐Castro et al. 2015, Lorrey et al. 2016, Thornton et al. 2018, Sanderson 2018, Silva 2020, O’Connor et al. 2021–2.
The secondary literature on Molesworth is limited to Ormerod 1880. ‘Brusqueness and originality’ is from p. viii, ‘very kind to the poor’ p. viii, and ‘Miss Molesworth’s labours will not have been … useless’ pp. xxi–xxii. Her journals themselves are online at https://digital.nmla.metoffice.gov.uk/IO_3246bc57-dc56-4ac9-a372-9e5e439a3768/.
Interlude: Number symbols
Menninger 1969, Flegg 1983, Guitel 1975 and (compendiously) Ifrah 1998 discuss the evidence for number symbols; Chrisomalis 2014 is a helpful discussion of what we do not know; the sketch of a classification in this section relies on Chrisomalis 2010b (pp. 9–14). ‘There is no ideal numerical notation system’ is from Chrisomalis 2010b, p. 19.; ‘we do not stand at the end point’ from p. 434.
6 Machines that count: Around East Asia
Hong Gongshou: Counting with rods
Background on China’s history is from Mote and Twitchett 2008; on its languages CEWAL (chapter 41 by Peyraube) and writing systems WWS section 14 (by Boltz). Numeral notations are discussed in Chrisomalis 2010b (chapter 8); see also Chemla and Ma 2015 and Chemla 2019.
On Ming-period government and economy I have used Mote and Twitchett 2008 chapter 1 (Hucker) and Ma and von Glahn 2022 chapter 9 (Lamouroux and von Glahn); on taxation in the period Huang 1974 and Mote and Twitchett 2008 chapter 2 (Huang), and on the census registers in particular Rhee 2005 and Zhang 2008. The source quoted here is from Rhee 2005; ‘the relocation of a particular business tax station’ and ‘as late as 1578 the imperial university’ are from Huang 1974, pp. 176 and 5 respectively. ‘A labor service payment of 0.0147445814487 taels of silver’ and ‘provided a paradise for lower-echelon tax collectors’ are both from Mote and Twitchett 2008 chapter 2 (Huang), p. 149.
Discussions of the counting rods and rod numerals are found in Volkov 1994, 2002, 2018; the classic Chinese mathematical texts are also discussed in Libbrecht 1973 (p. 447 for ‘Farmer A makes over 407 mou’), Martzloff 1997, OHHM chapter 7.1 (Cullen), Katz 2007 (chapter 3 by Dauben), Dauben 2019 and Yiwen 2020. ‘Units are vertical, tens are horizontal’ is quoted in Chrisomalis 2010b, p. 264; ‘could move his counting-rods as if they were flying’ is from Needham and Wang 1959, p. 72; ‘the method is complicated’ in Katz 2007 (chapter 3 by Dauben), p. 376.
Kiyoshi Matsuzaki (and Thomas Wood): Counting with beads
Primary sources for the contest are Anon 1946 and Kojima 1954; the step backwards for ‘the machine age’ and ‘“civilization” had “tottered”’ are both quoted on p. 12. Further background information is taken from Duus 2008 esp. chapters 4 (Fukui), 7 (Coox), 10 (Kōsai and Goble); Chrisomalis 2020 (p. 74) mentions the earlier tradition of such contests. Another important primary source for soroban use in the mid-twentieth century is Kojima 1963; useful histories include Pullan 1968 and Schärlig 2006. ‘Children go there after school’ is from Stigler et al. 1986, p. 454.
‘The character “Suan”’ is from Dauben 1998, p. 1348 note 12. Cognitive aspects of the suanpan and soroban are discussed in Hatano et al. 1977, Stigler et al. 1986, Hatta et al. 1989, Hishitani 1990, Miller and Stigler 1991, Yujie et al. 2020 and Wang 2020. Related but more general issues in visual processing are reported in Alvarez and Cavanagh 2005, Halberda et al. 2006 and Feigenson 2008. More specifically on the mental abacus, see Stigler et al. 1982, Stigler 1982, 1984, Hatano and Osawa 1983, Hatano et al. 1987, Frank and Barner 2011, Kim 2016, Barner et al. 2016, Cho and So 2018, Brooks et al. 2018 and Cheng et al. 2021.
Hermann Hollerith and Kawaguchi Ichitaro: Counting machines
Kawaguchi’s Electric Tabulation Machine is pictured and described at https://museum.ipsj.or.jp/en/computer/dawn/0056.html and https://museum.ipsj.or.jp/en/heritage/denkishukeiki.html. On the Japanese census see Matsuda 1981.
On the 1890 US census, important primary sources are Walker 1888, Wright 1889, Anon 1890 (‘The blanks which had been filled up’ is on p. 132) and Porter 1891, 1894. Historical reflections from soon after that census include Hunt 1899, Wright 1900 (pp. 69–79), SHCA 1903 (see pp. 173–4) and Wilcox 1914. ‘A very tidy and airy machineshop’ and ‘your tympanums all tingle’ are quoted in Austrian 1982, pp. 60–1, respectively from the Washington Star of 26 June 1890 and the Saginaw (Michigan) Weekly of 23 October 1890.
‘The unsettled area’ of the American continent is from Turner 1986, p. 1. Modern histories of the US census I have used include Anderson 1988, USCB 2002, Schor 2017 and Ruggles and Magnuson 2020. The loss of the 1890 records is discussed in O’Mahony 1991, Blake 1996 and Dorman 2008; ‘might every one of them be burned up’ is from Martin 1891, p. 525.
Hollerith’s own descriptions of his inventions appear in Hollerith 1889a, 1889b, 1894 among many other sources. Other (relatively) technical discussions from the period are Martin 1891, Newcomb 1913 (adoption by ‘nearly every civilized country’ on p. 82) and Felt 1916. From the voluminous more recent secondary literature I have used Truesdell 1965, Bashe 1986 (with one of the fullest descriptions of how the system actually worked), Reid-Green 1989, Campbell-Kelly 1989, Norberg 1990, Kistermann and Reston 1991, USCB 1991 and Campbell-Kelly et al. 2014. Additionally on Hollerith himself I have used Austrian 1982 and American National Biography (https://www.anb.org).
On the later development of the punched card I have consulted Lubar 2004 and Heide 2008, 2009.
Sia Yoon: Counting likes
On the history of computers I have used Akera and Nebeker 2002, Haigh and Ceruzzi 2021, O’Regan 2008 and Campbell-Kelly et al. 2014; on the digital revolution Hobart and Schiffman 1998, Deuze 2006, Gere 2008, Clarke 2012 and George 2020. ‘Consensual hallucination’ is from Neuromancer (William Gibson, 1984), p. 5.
On Korean history I use Connor 2002, Kim 2017 (especially Zur’s chapter) and Seth 2020; on technology therein Choi 2017. On K-pop and Korean Manga, Lie 2014, Pasfield-Neofitou et al. 2016 and Jin 2016, and on the webtoon phenomenon Jin 2015 (‘on a bustling subway ride to work’ is from p. 193), Kim and Yu 2019, Yecies et al. 2020, Shim et al. 2020 and Cho 2021. Sia’s story is online at https://www.webtoons.com/en/drama/my-dud-to-stud-boyfriend/list?title_no=4353.
7
Counting words and more in the Pacific world
Ayangkidarrba: Counting eggs
On the Australian archaeology I have used CWP chapters 1.35 (White) and 1.37 (Bird), and Cochrane and Hunt 2018 (chapters by Cochrane and Hunt, and O’Connor). An important recent publication on the earliest archaeology in the continent is Clarkson et al. 2017, with accompanying comments and reply (Clarkson et al. 2018).
A key anthropological source for this chapter is Keen 2004. On the languages of the continent it is important to be aware on the one hand of Dixon 1980, 2002, and on the other of such accounts as Bowern and Koch 2004, Campbell and Poser 2008 (pp. 318–25), and CWP chapter 1.42 (Heggarty and Renfrew). Additional information is taken from Hale 1975, Evans 2003, Cunningham et al. 2006 (chapters 5 and 12), Miyaoka et al. 2007 (chapter 12 by Walsh) and Leitner and Malcolm 2007. On counting in Australian languages I use Harris 1982, 1987, Sayers 1982, McRoberts 1990, Bowern and Zentz 2012, Epps et al. 2012, Treacy et al. 2015 and Zhou and Bowern 2015.
Specifically on Groote Eylandt and the Anindilyakwa I have used Worsley 1954, Turner 1974 (‘would awake at sunrise’: p. 165), and for the contact with Macassan people May et al. 2009. On their language, Leeding 1989 and van Egmond and Baker 2020. On their counting Stokes 1982, which I follow, describing the situation of number words and counting documented there (including rejecting a Macassan source for the lexemes and the base 5 system). ‘The hand is held loosely’ and ‘there is a story on Groote Eylandt’ are from Stokes 1982, pp. 43 and 39. Butterworth et al. 2008 gives a different account of the number words in this language and states that they are not (now) taught to children, presumably reflecting recent change.
Oksapmin: body count
For the context of this chapter in the ethnography and history of Papua New Guinea I have consulted Brongersma and Venema 1962 and Gammage 1998, and on the languages of the region the remarks in CWP chapter 1.42 (Heggarty and Renfrew), Dixon 2002 and Campbell and Poser 2008, as well as (particularly) Lean 1992 and Owens et al. 2018. On the Oksapmin in particular I have used Perey 1973 (p. 27 for ‘a raft crossing described as “very risky”’). On the Oksapmin language, see also Loughnane and Fedden 2011.
On counting in New Guinea, the classic and monumental studies are Lean 1992 and Owens 2001, synthesised in Owens et al. 2018. I have also referred to Laycock 1975, De Vries 2014 and Dwyer and Minnegal 2016, and to the classic Mimica 1988. On the cognitive and educational side, Matang 2005 and Matang and Owens 2014 are valuable.
