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Reinventing machines: the transmission history of the Leibniz calculator


This paper argues that we should take into account the process of historical transmission to enrich our understanding of material culture. More specifically, I want to show how the rewriting of history and the invention of tradition impact material objects and our beliefs about them. I focus here on the transmission history of the mechanical calculator invented by the German savant Gottfried Wilhelm Leibniz. Leibniz repeatedly described his machine as functional and wonderfully useful, but in reality it was never finished and didn't fully work. Its internal structure also remained unknown. In 1879, however, the machine re-emerged and was reinvented as the origin of all later calculating machines based on the stepped drum, to protect the priority of the German Leibniz against the Frenchman Thomas de Colmar as the father of mechanical calculation. The calculator was later replicated to demonstrate that it could function ‘after all’, in an effort to deepen this narrative and further enhance Leibniz's computing acumen.

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1 See Napier John, Mirifici logarithmorum canonis descriptio, Edinburgh: Ex officinâ Andreæ Hart bibliopôlæ, 1614; also Napier, Rabdologiæ, Seu Numerationis per Virgulas Libri Duo Cum Appendice de Expeditissimo Multiplicationis Promptuari, Edinburgh: Excudebat Andreas Hart, 1617. For Morland see Morland Samuel and Adamson Humphry, A New and Most Useful Instrument for Addition and Subtraction of Pounds, Shillings, Pence, and Farthings: Without Charging the Memory, Disturbing the Mind, or Exposing the Operator to Any Uncertainty : Which No Method Hitherto Published, Can Justly Pretend to, London?: s.n., 1672; Morland Samuel, The Description and Use of Two Arithmetick Instruments: Together with a Short Treatise, Explaining and Demonstrating the Ordinary Operations of Arithmetick : As Likewise a Perpetual Almanack and Several Useful Tables: Presented to … Charles II …, London: Printed and to be sold by Moses Pitt …, 1673. The most recent work on Morland is Ratcliff J.R., ‘Samuel Morland and his calculating machines c.1666: the early career of a courtier–inventor in Restoration London’, BJHS (2007) 40(2), pp. 159179. Some of these aids were also made of brass.

2 Pascal Blaise, ‘Privilege de la machine arithmetique’ in Pascal, Oeuvres complètes, vol. 2, Paris: Desclée de Brouwer, 1970, pp. 713714, 714.

3 For the broader context of Leibniz's life at this time see Antognazza Maria Rosa, Leibniz: An Intellectual Biography, Cambridge: Cambridge University Press, 2009, esp. pp. 79123. Leibniz procured himself during this time a copy of Pascal's Pensées. This was probably the source of his knowledge about the Pascaline. Gottfried Wilhelm Leibniz et al., Sämtliche Schriften und Briefe, Berlin etc. (Conventionally known as Akademie Ausgabe, here abbreviated AA), I, 1, p. 436.

4 Leibniz based his assumptions on Carcavy's suggestion that he could obtain access to Colbert if he could produce what he called a ‘chose réel et solide’. AA (2006 edn) II, 1, p. 308.

5 Carcavy's invitation is recorded in AA (2006 edn), II, 1, p. 208. Evidence of Leibniz's visit in the spring of 1672 is a letter from Carcavy saying, ‘Vous pouvez Monsieur, venir quand il vous plaira, mais si vous jugez à propos que ce soit demain du matin je vous attendray, Et ne vous donnerois pas cette peyne si je n'avois des affaires importantes qui m'arrestent a la maison.’ AA II, 1, p. 339. When he saw the Pascaline, Leibniz made notes on its design. See Leibniz manuscripts in the Hannover Library XLII, 5, p. 5 (Leibniz Handschriften – here abbreviated LH).

6 There is currently little work on this topic. The most recent incursion is Jones Matthew L., ‘Improvement for profit: calculating machines and the prehistory of intellectual property’, in Riskin Jessica and Biagioli Mario (eds.), Nature Engaged: Science in Practice from the Renaissance to the Present, Basingstoke: Palgrave Macmillan, 2012, pp. 125147. See also Jones, Reckoning with Matter: Calculating Machines, Innovation, and Thinking about Thinking from Pascal to Babbage, Chicago: The University of Chicago Press, forthcoming 2014. In German see Ludolf von Mackensen, ‘Die Vorgeschichte und die Entstehung der 4-Spezies-Rechenmaschine von Gottfried Wilhelm Leibniz nach bisher unerschlossenen Manuskripten und Zeichnungen mit einem Quellenanhang der Hauptdokumente’, PhD dissertation, University of Munich, 1968. This dissertation covers the Paris phase of the development history of the machine, particularly the period 1671–1677. The interpretative line that Mackensen takes is centered on Leibniz's visionary genius, but the dissertation provides important factual information for the period covered. The addendum contains transcriptions of several manuscripts. For the period of time the Leibiz calculator was in Helmstedt under the supervision of Christian Wagner see Wilberg Ernst Eberhard, Die Leibniz'sche Rechenmaschine und die Julius-Universität in Helmstedt, Braunschweig: Universitätbibliothek der Technischen Universität Carolo-Wilhelmina, 1977. For the same period but focused on the artisans see Scheel Günter, ‘Helmstedt als Werkstatt für die Vervollkommnung der von Leibniz erfundenen und konstruierten Rechenmaschine (1700–1711)’, Braunschweigisches Jahrbuch für Landesgeschichte (2001) 82, pp. 105118.

7 J. A. Bennett, curator and historian of science, has been a tireless proponent for collapsing such distinctions. See, for example, Bennett J.A., ‘The mechanics’ philosophy and the mechanical philosophy’, History of Science (1986) 24, pp. 128.

8 The origin of these ideas is Thomas Kuhn's 1962 paper ‘The historical structure of scientific discovery’, Science (1962) 136, pp. 760–764. Rheinberger's is the most recent take on the subject. See Rheinberger Hans-Jörg, ‘Experimental systems: historiality, narration, and deconstruction’, Science in Context (1994) 7, pp. 6581. See also Schaffer Simon, ‘Scientific discoveries and the end of natural philosophy’, Social Studies of Science (1986) 16, pp. 387420. For other similar work see Woolgar S.W., ‘Writing an intellectual history of scientific development: the use of discovery accounts’, Social Studies of Science (1976) 6, pp. 395422. And a compelling case study is Brannigan Augustine, ‘The reification of Mendel’, Social Studies of Science (1979) 9, pp. 423454.

9 Rheinberger, op. cit. (8), p. 412.

10 According to Mackensen, op. cit. (6), pp. 31–51 and transcriptions 128 ff., the earliest manuscripts from the Mainz period are ‘Instrumentum Arithmeticum and Instrumentum panarithmeticon oder die “Lebendige Rechenbank”’. Cf. LH XLII, 5, pp. 15–17. The correct order of the ‘Instrumentum arithmeticum’ manuscript is pp. 16v, 16r, 15v, 15r.

11 This is one of the more interesting justifications Leibniz found for the machine. The full citation is as follows: ‘La machine dont nous donnons icy la description exterieure, fait voir que l'esprit humaine peut trouver le moyen de se transplanter de telle façon dans une matière inanimée qu'il luy donne le pouvoir, de faire bien plus qu'il n'auroit fait lui-même : pour convaincre sensiblement ceux qui ont de la difficulté à concevoir comment le Createur puisse longer une apparence d'esprit un peu plus générale dans les corps des bestes qoyque fourni de tant d'organes ; puisque le loton même a peu recevoir l'imitation d'une opération de la raison qui est à la vérité particulière ou determinée, mais des plus difficiles, veu que les pythagoriciens croyaient de pouvoir par là distinguer l'homme de la beste et de faire entrer dans sa définition la faculté de se servir de nombres.’ LH XLII, 5, II, p. 33r.

12 LH XLII, 5, pp. 1–4, also 67–69. Transcribed in Jordan W., ‘Die Leibnizsche Rechenmaschine’, Zeitschrift für Vermessungswesen (1897) 26, pp. 289315.

13 Mackensen, op. cit. (6), pp. 143–162. and LH XLII, V, pp. 48–53 and 54–57.

14 Was Leibniz rewriting the history of his own machine? Leibniz mentions the roue a dents inégales in the 1677 letter to the Parisian clockmaker Olivier. Cf. Mémoire pour monsieur Olivier touchant la machine perfectionée, LH XLII, 4, II, p. 7. The undated manuscript ‘Machina aritmetica qualem curavi elaborarem’ seems to be the first comprehensive description of the wheel with unequal teeth. See LH XLII, 4, II, pp. 11–13. This is presented as a correction of and improvement over a previous technology.

15 See letters to Carcavy and correspondence with Ferrand from that year. AA I, 1, p. 452. In January 1673 the so-called ebauche was shown to the members of the Royal Society. See Birch Thomas, The History of the Royal Society, vol. 2, Hildesheim: Georg Olms, 1968, p. 87. Back in Paris the development work continued while Leibniz was applying for funding from the Académie royale des sciences to bring his idea to the market. See Letter to the Académie, in Mackensen, op. cit. (6), p. 174. By 1675, having exhausted all of his possibilities in Paris, he accepted the position of councilor and librarian to the Duke of Hanover and made his way back to the German states.

16 Hansen's letter of May 1679, AA I, 2, p. 468.

17 Leibniz's manuscripts from this period are mainly focused on this. See, for example, the paper titled ‘Machine Arithmetique’ 8 Maii 1682, LH XLII, 4, II, p. 40; also ‘Ma machine Arithmetique Juinius 1684’, LH XLII, 4, II, p. 41.

18 See correspondence with Kölbing, first letter, AA I, 12, N. 65.

19 Leibniz added a pentagonal wheel to each carry to show whether the carry has been correctly performed. See AA I 17, 105. There is speculation that this would have allowed manual intervention to correct errors. See Stein Erwin and Kopp Franz Otto, ‘Konstruktion und Theorie der Leibnizschen Rechenmaschinen im Kontext der Vorläufer, Weiterentwicklungen und Nachbauten: Mit einem Überblick zur Geschichte der Zahlensysteme und Rechenhilfsmittel’, Studia Leibnitiana: StL; Zeitschrift für Geschichte der Philosophie und der Wissenschaften (2010) 42(1), pp. 1128.

20 AA I, 17, p. 11.

21 The correspondence with Wagner is still largely in manuscript. It was kept at the university of Halle. See Halle UB YG 13; also AA I, 19. Günter Scheel has done research to uncover the names of the artisans who worked on the machine. Scheel, op. cit. (6).

22 Three clockmakers were employed there: the two brothers of the mathematician Mathias Has and another unknown artisan. The correspondence sent by Leibniz to Teuber was edited in August Nobbe, Godofredi Guilielmi L. B. de Leibnitz Lipsiensis Epistolae XLVI, 1845. For Teuber's letters to Leibiz see manuscripts at the Leibniz library in Hanover, Lbr. 916. The correspondence with Buchta is currently being edited. Some of the transcriptions are available online. See, accessed 10 February 2014.

23 This can be inferred from the correspondences, Teuber's testimonial and also the more recent replication attempts I discuss bellow. Contrary to what twenty-first-century replicators claim, the functioning replicas with modern materials actually show the limitations of Leibniz's machine at the time it was constructed. Even if the machine had used the manual correcting mechanism, as Stein has argued, it is unclear how well the historical machine would have worked and if it could actually work up to twelve digits as Leibniz claimed.

24 This appears in the correspondence with Teuber explaining the failure to make the carry mechanism work properly: ‘Ut decadicarum rotarum translatio bene procedat, necesse est (uti nemo Te iudicare poterit) tam ipsas decadicas rotas, quam fundamentales illas ita aequaliter collocates esse, ut quaevis decadica cuivis fundamentali accomodari posit, quod diligentiam et accurationem in opifice desiderat, et velut lapis est Lydius, quo quid valeat dignosci poterit. Quanto, obsecro, res future esset difficilior; si etiam, ut olim tot aliae particulae, quaevis tot aliis accommodari deberent? Desunt nobis »in Germania« homines satis periti et industrii, qui bene concepta bene exsequantur.’ 15 August 1714, Nobbe, op. cit. (22), p. 17. See also Shapin Steven, A Social History of Truth: Civility and Science in Seventeenth-Century England, Chicago: The University of Chicago Press, 1999, pp. 355407.

25 A good example is the letter to the Académie des sciences. See Mackensen, op. cit. (6), p. 178; also correspondence with Oldenburg, AA, III, p. 1; and correspondence with Arnauld, Tchirnhaus and Huygens.

26 AA II, 1, 164.

27 AA I, 1, 488.

28 Birch, op. cit. (15), p. 87.

29 AA III, 1, 118–119.

30 ‘Machinulam Tuam arithmeticam, quam perfecisse Te antehac jam significasti, lubentes equidem lustraremus, si promissi Tui, Soc. Regiae in public congress facti, memor occasione commode transmitter eam velles.’ AA, III, I, p. 172. Cf. commentator's note on p. 181 in the volume for the significance of ‘machinulam’ in this context. He also broke a similar promise to the Académie des sciences in Paris, where he anticipated he would send models of the machine to the Observatoire astronomique and other Parisian institutions.

31 AA III, 1, p. 283.

32 The relevant letters are at AA III, 6, pp. 271, 278, 305.

33 Leibniz himself had sent the book over to help Haes identify an instrument his patron purchased in England. AA III, 6, p. 368.

34 Gottfried Wilhelm Leibniz, ‘Brevis descriptio machinae arithmeticae’, Miscellanea Berolinensia Ad Incrementum Scientiarum (1710) 1.

35 Leupold Jacob, Theatrum Arithmetico-Geometricum, Leipzig: Gleditsch, 1726.

36 Matthew Jones interprets this as evidence for emulation. Cf. Jones, Reckoning with Matter, op. cit. (6).

37 The letters from Kästner are found together with the manuscripts pertaining to the machine. LH XLII, 4, p. 31r.

38 Kästner Abraham Gotthelf, Fortsetzung der Rechenkunst in Anwendungen auf mancherley Geschäffte, Göttingen: Vandenhoeck, 1786, p. 582 f.

39 Bischoff Johann Paul, Versuch einer Geschichte der Rechenmaschine, Ansbach, 1804, p. 125.

40 A drawing of Hahn's stepped wheel is found in Bischoff's book.

41 Kistermann F.W., ‘When Could Anyone Have Seen Leibniz's Stepped Wheel?’, IEEE Annals of the History of Computing (1999) 21(2), pp. 6872, 70.

42 Jones, Reckoning with Matter, op. cit. (6).

43 Martin Ernst, The Calculating Machines (Die Rechenmaschinen): Their History and Development, Cambridge, MA: MIT Press, 1992, p. 41.

44 Gerke Rudolf, ‘Die Leibnizsche Rechenmaschine’, Zeitschrift für Vermessungswesen (1880) 9, pp. 305312.

45 Translation of the passage in full: ‘The thing was so well built that according to Leibniz and the testimony of other mathematicians, even a child could calculate with the machine. The whole world was filled with wonder over the invention. The great mathematicians such as Huygens, Thevenot, Arnaud and others paid Leibniz their homage and even the friends and relatives of Pascal had to admit the originality of the invention.’ Gerke, op. cit. (44), p. 307.

46 Gerke, op. cit. (44), pp. 310–311.

47 Jordan Wilhelm, Handbuch der Vermessungskunde, Stuttgart: Metzler, 1895, p. 136.

48 Jordan, op. cit. (47), p. 137.

49 Jordan's note at the end of Burkhardt's report. See Burkhardt A., ‘Die Leibnizsche Rechenmaschine’, Zeitschrift für Vermessungswesen (1897) 26, pp. 392398, 398.

50 Burkhardt, op. cit. (49), p. 395.

51 Burkhardt, op. cit. (49), p. 398.

52 Burkhardt, op. cit. (49), p. 393.

54 In addition to these two projects there is a third by Erwin Stein. It does not differ substantially, however, from Badur's and Lehmann's, being also an attempt to show that the machine worked ‘after all’. See Kopp Franz Otto and Stein Erwin, ‘Es funktioniert doch – mit zwei Korrekturen: Zehnerüberträge der Leibniz-Maschine; zur Konstruktion des hannoverschen Nachbaus der Vier-Spezies-Rechenmaschine von Leibniz’, Leibniz: Auf den Spuren des großen DenkersUnimagazin (2006) 3–4, pp. 5659; Kopp and Stein, ‘Konstruktive Verbesserungen im hannoverschen Modell der Leibnizschen Vier-Spezies-Rechenmaschine’, Einheit in der Vielheit, Teil 1, Hannover: Gottfried-Wilhelm-Leibniz-Ges, 2006, pp. 390–397; Stein Erwin and Kopp Franz Otto, ‘Calculemus! Neue hannoversche Funktionsmodelle zu den Rechenmaschinen von Leibniz’, Leibniz: auf den Spuren des großen DenkersUnimagazin (2006) 3–4, pp. 6063. The description of another reconstruction of a Leibniz cypher machine can be found in Rescher Nicholas, Leibniz and Cryptography: An Account on the Occasion of the Initial Exhibition of the Reconstruction of Leibniz's Cipher Machine, Pittsburgh, PA: University Library System, University of Pittsburgh, 2012.

55 Lehmann Nikolaus Joachim, ‘Leibniz als Erfinder und Konstrukteur von Rechenmaschinen’, in Nowak Kurt and Power Hans (eds.), Wissenschaft und Weltgestaltung, Hildesheim, 1999, pp. 255267.

56 The passage in full: ‘While the construction of the Dresden replica was progressing its functionality was studied in greater detail and it was recognized that the basic measurements and order of the parts allowed for the fully automatic carry of the decimals, which Leibniz wanted. Burkhardt followed in his restoration a theory that was based on false assumptions and wrongly modified the machine according to it.’ Lehmann, op. cit. (55), p. 176.

57 The passage in full reads, ‘The Dresden replica, which is capable of functioning and accords perfectly with Leibniz's intention, confirms perfectly the feasibility of his conception and also his ideas in great detail.’ Lehmann, op. cit. (55), p. 186.

58 Lehmann, op. cit. (55), p. 188.

59 Badur Klaus and Rottstedt Wolfgang, ‘Und sie rechnet doch richtig! Erfahrungen beim Nachbau einer Leibniz-Rechenmaschine’, Studia Leibnitiana (2004) 36, pp. 130146; Klaus Badur, ‘Neue Erkenntnisse zur Rechengenauigkeit der Leibniz Rechenmaschine: Erfahrungen mit einem originalgetreuen Nachbau’, in Einheit in der Vielheit, Vorträge, VIII, Hannover: Gottfried-Wilhelm-Leibniz-Ges, 2006, pp. 16–23.

60 Cf. Badur, op. cit. (59), 19.

61 He refers to the machine as an ‘original true replica with modern finishing techniques’ – ‘originalgetreue Nachbau mit moderner Fertigungstechnik’. Badur, op. cit. (59), 20.

62 The entire passage reads: ‘In conclusion I want to say that Leibniz was right when he wrote in his letter to Johann Bernouli on 26 May 1697 that he has a clear idea of the internal construction of the machine and that the perfection of the machine will depend only on the good implementation of his ideas. To this I want to add that the technical possibilities were not given at that time. Leibniz was ahead of his time with his ideas, relative to the implementation possibilities.’ Badur, op. cit. (59), 20.

63 Collins H.M., Changing Order: Replication and Induction in Scientific Practice, London: Sage Publications, 1985.

65 Mackensen, op.cit. (6), p. 44, suggests an analogy with a music box set-up depicted in Athanasius Kirchner's Musurgia universalis.

66 Hoyau M., ‘Description d'une machine à calculer nommée arithmomètre, de l'invention de M. Le Chevalier Thomas, de Colmar’, in Bulletin de la Société d'encouragement pour l'industrie nationale, Paris: Siège de la Société, 1822, pp. 356368, 357 – ‘l'aspect d'un escalier à marches très courtes, tournant autour d'une colonne’.

67 Hoyau, op. cit. (66). See also other documents available online at, accessed 11 February 2014.

68 See how the machine is presented in Archives des découvertes et des inventions nouvelles, Paris: Treuttel et Würtz, 1823, 279.

69 For example See Encyclopedia Britannica, at, accessed 11 February 2014.

This paper was a long time in the making and in the process I have incurred a number of debts. I wish to thank Mario Biagioli, Jimena Canales, Jeanne Peiffer, Siegmund Probst, Kapil Raj, Carsten Reinhardt, Suzanne Smith and Heidi Voskuhl, who saw versions of my manuscript and gave crucial feedback. The two anonymous reviewers have also provided detailed and immensely useful comments. I wish also to thank Matthew L. Jones, with whom I share a passion for calculating machines, for our fruitful exchange of manuscripts. I'm looking forward to the final publication of his innovative book on the history of calculating from Pascal to Babbage. Many thanks also to Professors Katharine Park and Shigehisa Kuriyama, who have supported my journey through graduate school in the friendliest, most generous and inspiring way possible. Finally, a special call-out to my friends Josh Freeman, Sungho Kimlee, Tom Wiesniewski, Ovidiu Stanciu and Radu Toderici.

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