Positive Thinking. Conceptions of Negative Quantities in the Netherlands and the Reception of Lacroix's Algebra Textbook
Revue d'histoire des mathématiques, Volume 6 (2000) no. 1, p. 95-126

The beginning of the 19th century witnessed the emergence of several new approaches to negative numbers. New notions of rigour made the 18th century conceptions of negative quantities unacceptable. This paper discusses theories of negative numbers emerging in the Netherlands in the early 19th century. Dutch mathematicians then opted for a different approach than that of their contemporaries, in Germany or France. The Dutch translation (1821) of Lacroix's Élémens d'algèbre illustrates the ‘Dutch' notion of rigour.

Au début du xixe siècle, des attitudes nouvelles par rapport aux nombres négatifs émergent. La notion de rigueur en mathématique se renouvelle, rendant inacceptables les approches qui s'étaient développées au xviiie siècle. Cet article présente les théories des nombres négatifs qui avaient cours aux Pays-bas au début du xixe siècle. Les mathématiciens néerlandais optaient pour une conception différente de celles de leurs contemporains en Allemagne et en France. La traduction néerlandaise des Élémens d'algèbre de Lacroix illustrera cette approche « néerlandaise » de la notion de rigueur.

@article{RHM_2000__6_1_95_0,
     author = {Beckers, Danny J.},
     title = {Positive Thinking. Conceptions of Negative Quantities in the Netherlands and the Reception of Lacroix's Algebra Textbook},
     journal = {Revue d'histoire des math\'ematiques},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {6},
     number = {1},
     year = {2000},
     pages = {95-126},
     zbl = {0965.01006},
     language = {en},
     url = {http://www.numdam.org/item/RHM_2000__6_1_95_0}
}
Beckers, Danny J. Positive Thinking. Conceptions of Negative Quantities in the Netherlands and the Reception of Lacroix's Algebra Textbook. Revue d'histoire des mathématiques, Volume 6 (2000) no. 1, pp. 95-126. http://www.numdam.org/item/RHM_2000__6_1_95_0/

[1] Alberts (Gerard), Atzema (Eisso), Maanen (Jan Van) [1999] Mathematics in the Netherlands. A brief survey with an emphasis on the relation to physics, 1560-1960, in: K.van Berkel, A.van der Helden, L.Palm, eds., A History of Science in the Netherlands, Leiden/Boston/Köln: Brill, 1999, pp.367-404. | MR 1686883

[2] Anonymous [1798] Iets over het denkbeeld van subtractie of aftrekking, met opzicht tot de algebra, Nieuwe Vaderlandsche Bibliotheek, 2 (1798), pp.466-470.

[3] Baayen (Pieter C.) [1978] ‘Wiskundig Genootschap' 1778-1978: some facts and figures concerning two centuries of the Dutch Mathematical Society, Nieuw Archief voor Wiskunde, (III) 26 (1978), pp.177-205. | MR 484104 | Zbl 0375.01001

[4] Badon Ghyben (Jacob), Strootman (H.) [1838] Beginselen der Stelkunst, Breda, 1838.

[5] Baudet (Pierre Joseph) [1820] Opleiding tot de kennis der algebra, Deventer, 1820-1830 [3 vols.; 3rd. edition 1842-1843].

[6] Baudet (Pierre Joseph) [1824] Meetkundig Schoolboek, Deventer, 1824.

[7] Beckers (Danny) [1996] Mathematics as a way of life - a biography of the mathematician Jacob de Gelder (1765-1848), Nieuw Archief voor Wiskunde, (IV) 14 (1996), pp.275-297. | MR 1402848 | Zbl 0856.01036

[8] Beckers (Danny) [1998a] Het is al Mathesis dat de klok slaat, De Negentiende Eeuw, 22 (1998), pp.220-234.

[9] Beckers (Danny) [1998b] J.F.L.Schröder on the foundations of geometry, Nieuw Archief voor Wiskunde, (IV) 16 (1998), pp.113-134. | MR 1645268 | Zbl 0966.01016

[10] Beckers (Danny) [1999] Lagrange in the Netherlands: Dutch attempts to obtain rigour in calculus, 1797-1840, Historia mathematica, 26 (1999), pp.224-238. | MR 1710337 | Zbl 0939.01006

[11] Bemmelen (Abraham Van) [1817] Lessen over de Algebra of Stelkunst, Den Haag, 1817-1818 [two volumes, 4th edition 1854].

[12] Blassière (Jean Jacques) [1770] Institution du calcul numérique et littéral, La Haye, 1770 [two volumes].

[13] Bockstaele (Paul P.) [1978] Mathematics in the Netherlands from 1750 to 1830, Janus, 65 (1978), pp.67-95. | MR 510914 | Zbl 0394.01009

[14] Booy (Engelina Petronella De), Engel (J.) [1985] Van Erfenis tot Studiebeurs, Delft: Maatschappij tot Nut van 't Algemeen, 1985.

[15] Buys Ballot (Christopher H.D.) [1852] Beginselen en Gronden der Meetkunde, Utrecht, 1852 [3rd. edition 1860].

[16] Carnot (Lazare Nicolas Marguerite) [1801] De la corrélation des figures de géométrie, Paris, 1801 [An IX].

[17] Carnot (Lazare Nicolas Marguerite) [1803] Géométrie de position, Paris, 1803 [An XI].

[18] Cerutti (Sophie) [1998] Illegale handel in boeken rond 1791, De Achttiende Eeuw, 30 (1998), pp.59-78

[19] Clairaut (Alexis Claude) [1760a] Beginzelen der Geometrie, Amsterdam, 1760 [translator: Arnoldus Bastiaan Strabbe].

[20] Clairaut (Alexis Claude) [1760b] Gronden der Algebra, Amsterdam, 1760 [translator: A.B.Strabbe].

[21] Condillac (Etienne Bonnot) [1981] La langue des calculs, Lille: Presses Universitaires de Lille, 1981 [original edition published posthumously: Paris, 1798]. | MR 648288

[22] Crummel (Joseph) [1776] Het Nut der Algebra, Purmerend, 1776 [translator: Jacob Oostwoud].

[23] Dhombres (Jean Et Nicole) [1997] Lazare Carnot, Paris: Fayard, 1997. | MR 924847

[24] Diesterweg (W.A.) [1831] Beiträge zu der Lehre von den positiven und negativen Grössen, Bonn, 1831.

[25] Euler (Leonhard) [1773] Volledige Inleiding tot de Algebra, Amsterdam, 1773 [2 volumes; translator unknown; second edition : Amsterdam, 1807, by M.I.S. Bevel].

[26] Ferraro (Giovanni) [1998] Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula, Historia mathematica, 25 (1998), pp.290-317. | MR 1649957 | Zbl 0917.01018

[27] Förstemann (Wilhelm A.) [1817] Über den Gegensatz positiver und negativer Grössen, Nordhausen, 1817.

[28] Frend (William) [1796] The Principles of Algebra, London, 1796-1799 [two volumes].

[29] Gelder (Jacob De) [1806a] Redevoering over de ware aart en voortreffelijkheid der wiskunst, Den Haag, 1806.

[30] Gelder (Jacob De) [1806b] Handleiding tot de beschouwende en werkdadige Meetkunst, Den Haag/ Amsterdam, 1806.

[31] Gelder (Jacob De) [1810] Beginselen der Meetkunst, Den Haag/Amsterdam, 1810 [revised several times, 4th edition 1850].

[32] Gelder (Jacob De) [1811] Meetkundige Analysis, Den Haag/Amsterdam, 1811-1813.

[33] Gelder (Jacob De) [1815] Proeve over den waren aard van den positieven en negatieven toestand der grootheden, Amsterdam, 1815.

[34] Gelder (Jacob De) [1823] Beginselen der differentiaal -, integraal - en variatierekening, Den Haag/ Amsterdam, 1823.

[35] Gelder (Jacob De) [1826] Allereerste Gronden der Stelkunst, Amsterdam/Den Haag, 1826 [third edition 1830].

[36] Gerritsen (Willem P.) [1997] Het Koninklijk Instituut (1808-1851) en de bevordering van wetenschap en kunst, Amsterdam: KNAW, 1997.

[37] Goudswaard (Nicolaas B.) [1981] Vijf-en-zestig jaren nijverheidsonderwijs, Assen: Van Gorcum, 1981.

[38] Grabiner (Judith V.) [1990] The Calculus as Algebra, New York et al.: Garland, 1990. | MR 1098713

[39] Grattan-Guinness (Ivor) [1997] The Fontana History of the Mathematical Sciences, Glasgow: Fontana Press, 1997. | MR 1601822 | Zbl 0876.01003

[40] Gregory (Frederick) [1983] Neo-Kantian foundations of geometry in the German romantic period, Historia mathematica, 10 (1983), pp.184-201. | MR 703943 | Zbl 0522.01006

[41] Halcken (Paul) [1767] Mathematisch Zinnen-Confect, Purmerend, 1767 [translator J.Oostwoud].

[42] Hammond (Nathaniel) [1759] De Algebra gemakkelyk gemaakt, Amsterdam, 1759 [translation of the second edition, London, 1752; translator unknown].

[43] Hiddinga (Gerloff Feddes) [1735] Aanleyding tot de Algebra of Stelkonst, Purmerend, 1735 [translation of: Anleitung zur Algebra speciosa, Hamburg, 1735; translator J. Oostwoud].

[44] Huguenin (Ulrich) [1817] Ontwikkeling van eenige Trigonometrische Formulen en Reeksen, Wiskundige Verhandelingen, I (1817), pp.1-17.

[45] Janssen (Jan A.M.M.) [1989] Op weg naar Breda, Den Haag: Koninklijke Landmacht, 1989.

[46] Kool (Marjolein) [1999] Die conste vanden getale, Hilversum: Verloren, 1999.

[47] Lacroix (Sylvestre François), [1797] Élémens d'algèbre, Paris, 1797 [all quotes from 11th edition, 1815].

[48] Lacroix (Sylvestre François), [1801] Complément des élémens d'algèbre, Paris, 1801 [2nd edition].

[49] Lacroix (Sylvestre François), [1808] Traité élémentaire d'arithmétique, Paris, 1808 [8th edition].

[50] Lacroix (Sylvestre François), [1811] Anfangsgründe der Algebra, Mainz, 1811 [translated by M.Metternich].

[51] Lacroix (Sylvestre François), [1821a] Beginselen der Stelkunst, Den Haag/Amsterdam, 1821 [translated by Isaak Riewert Schmidt; 2nd edition 1825].

[52] Lacroix (Sylvestre François), [1821b] Grondbeginselen der Beschrijvende Meetkunst, Amsterdam/Den Haag, 1821 [translated by I.R.Schmidt; 2nd edition 1833].

[53] Lacroix (Sylvestre François), [1821c] Beginselen der meetkunst, Amsterdam/Den Haag, 1821 [translated by I.R.Schmidt; 3rd edition 1838].

[54] Lacroix (Sylvestre François), [1822] Beginselen der goniometrie en trigonometrie, Den Haag/Amsterdam, 1822 [translated by I.R.Schmidt; 5th edition 1856].

[55] Martin (Benjamin) [1763] Algemeene Oeffenschoole van Konsten en Weetenschappen, Amsterdam, 1763 [translation of the encyclopedic work; translator unknown].

[56] Maseres (Francis) [1758] Dissertation on the Use of the Negative Sign in Algebra, London, 1758

[57] Maseres (Francis) [1800] Tracts on the Resolution of Affected Algebraick Equations, London, 1800

[58] Mauduit (Antoine R.) [1763] Inleiding tot de Keegelsneeden, Den Haag, 1763 [reprinted 1778; translated by J.J.Blassière].

[59] Nieuwland (Pieter) [1789] Redevoering benevens Aanspraak, Amsterdam, 1789.

[60] Peacock (George) [1830] A Treatise on Algebra, London, 1830 [2 volumes; the 1842 edition of this book has been reprinted in New York: Scripta Mathematica, 1940].

[61] Peacock (George) [1833] Report on the recent progress and present state of certain branches of analysis, Report of the Meeting of the British Association for the Advancement of Science, 3 (1833), pp.185-253.

[62] Pycior (Helena M.) [1981] George Peacock and the British origins of symbolical algebra, Historia mathematica, 8 (1981), pp.23-45. | MR 618162 | Zbl 0458.01007

[63] Pycior (Helena M.) [1982a] Early criticism of the symbolical approach to algebra, Historia mathematica, 9 (1982), pp.392-412. | MR 679222 | Zbl 0501.01007

[64] Pycior (Helena M.) [1982b] Historical roots of confusion among beginning algebra students: a newly discovered manuscript, Mathematics Magazine, 55 (1982), pp.150-156. | MR 1572420 | Zbl 0497.01015

[65] Pycior (Helena M.) [1987] British abstract algebra, in Grattan-Guinness (Ivor), ed., History in Mathematics Education, Paris: Société française d'histoire et de philosophie des sciences, 1987, pp.152-168. | Zbl 0658.01008

[66] Pycior (Helena M.) [1997] Symbols, Impossible Numbers and Geometric Entanglements, Cambridge: Cambridge University Press (1997) | MR 1470715 | Zbl 0876.01002

[67] Quetelet (Lambert Adolphe Jacques) [1826] Review of [De Gelder 1823], Correspondance mathématique et physique, II (1826), pp.244-245.

[68] Rees (Richard Van) [1825] Review of [De Gelder 1815], Correspondance mathématique et physique, I (1825), pp.290-296.

[69] Schmidt (Isaak Riewert) [1817] Verhandeling over eenige eigenschappen van den regtlijnigen driehoek, Wiskundige Verhandelingen van het Genootschap te Amsterdam, I (1817), pp.18-32.

[70] Schmidt (Isaak Riewert) [1822a] Beginselen der Differentiaal- en Integraal Rekening. Ten gebruike van de kadetten der koninklyke Artillerie- en Genieschool te Delft., Den Haag/Amsterdam, 1822 [2nd edition 1837].

[71] Schmidt (Isaak Riewert) [1822b] Beginselen der Hoogere Meetkunst, Den Haag/Amsterdam, 1822 [2nd edition 1826].

[72] Schmidt (Isaak Riewert) [1823] Beginselen der statica, Den Haag/Amsterdam, 1823-1824 [2 volumes].

[73] Schmidt (Isaak Riewert) [1825] Beginselen der dynamica, Den Haag/Amsterdam, 1825.

[74] Schmidt (Isaak Riewert) [1827] Nagelaten Wiskundige Verhandelingen, Den Haag/Amsterdam, 1827.

[75] Schröder (Johann F.L.) [Notes] Lecture Notes of a course by Schröder, written by a(n anonymous) student in 1816; University of Utrecht Archival Collection, inv.nr. VIII J 15.

[76] Schröder (Johann F.L.) [1831] Elementa Matheseos Purae, Utrecht, 1831-1834 [two volumes].

[77] Schubring (Gert) [1982] Die Mathematik an der École normale des Jahres III, in Schmithals (F.), ed., Wissen und Bewußtsein, Studien zur Wissenschaftsdidaktik der Disziplinen, Hamburg: AHD, 1982, pp.103-133. | Zbl 0516.01018

[78] Schubring (Gert) [1986] Ruptures dans le statut mathématique des nombres négatifs, Petit x, 12 (1986), pp.5-32. | MR 910869

[79] Schubring (Gert) [1988] Un savant des Lumières - Un livre élémentaire pour la république, in: Condorcet. Moyens d'apprendre à compter sûrement et avec facilité, Paris: ACL Éditions, 1988, pp.157-175.

[80] Schubring (Gert) [1991] Spezialschulmodell versus Universitätsmodell: die Institutionalisierung von Forschung, in Schubring (Gert), ed., Einsamkeit und Freiheit neu besichtigt, Stuttgart: Steiner, 1991, pp.276-326.

[81] Schubring (Gert) [1996] Changing cultural and epistemological views on mathematics and different institutional contexts in nineteenth-century Europe, in Goldstein (Catherine), Gray (Jeremy) and Ritter (James), eds, L'Europe mathématique, Mythes, histoires, identités, Paris: Éditions de la Maison des sciences de l'homme, 1996, pp.362-388. | MR 1770136 | Zbl 0880.01013

[82] Simpson (Thomas) [1770] Gronden der Meetkonst, Amsterdam, 1770 [translator was Arnoldus Bastiaan Strabbe; second edition published in 1790].

[83] Smeur (A.J.E.M.) [1978] The rule of false applied to the quadratic equation, in three sixteenth century arithmetics, Archives internationales d'histoire des sciences, 28 (1978), pp.66-101. | MR 553610 | Zbl 0385.01014

[84] Smid (Harm Jan) [1997] Een onbekookte Nieuwigheid, Delft: Delft University Press, 1997.

[85] Smid (Harm Jan) [1998] Wiskundeonderwijs in de negentiende eeuw. Een omstreden succesverhaal, De negentiende eeuw, 22 (1998), pp.209-220.

[86] Steenstra (Pibo) [1763] Beginselen der Meetkunst, Leiden, 1763 [10th edition 1835].

[87] Swinden (Jan Hendrik Van) [1770] Verhandeling behelzende eene nieuwe betooginge van de verheffing der Grootheid a±b tot de onbepaalde magt n, Verhandelingen Hollandsche Maatschappy der Weetenschappen, 12 (1770), pp.334-358.

[88] Swinden (Jan Hendrik Van) [1790] Grondbeginsels der Meetkunde, Amsterdam, 1790 [from a Latin textbook from 1786; second edition 1816; German translation appeared in 1834].

[89] Swinden (Jan Hendrik Van) [Notes] Lecture Notes of a course by Van Swinden, written by one of his students (A.J. Deiman), ca. 1800; University of Utrecht Archival Collection, inv.nr. VIII G 12.

[90] Tholen (Jacobus Pierson) [1784] Theses philosophicæ, Franeker, 1784.

[91] Venema (Pieter) [1714] Een korte en klare Onderwysinge in de beginselen der Algebra ofte Stelkonst, Groningen, 1714 [seventh edition Amsterdam, 1803].

[92] Wettengel (Michael) [1990] Die Geschichte der wissenschaftlichen Gesellschaften in Hamburg unter besonderer Berücksichtigung der Mathematischen Gesellschaft in Hamburg von 1690, in Barlotti (A.), ed., Festschrift der Mathematischen Gesellschaft in Hamburg zu ihrem 300jährigen Bestehen 1990, Hamburg: Mathematische Gesellschaft, 1990, pp.61-205 [=Mitteilungen der Mathematischen Gesellschaft in Hamburg 12 Heft 1]. | MR 1069182 | Zbl 0703.01018

[93] Wolff (Christian) [1738] Grond-Beginzelen van Alle de Mathematische Wetenschappen, Amsterdam, 1738-1739 [six volumes; translated by J.C. van Sprögel, reprinted in 1758].

[94] Wolff (Christian) [1740] Volkoomen Wiskundig Woordenboek, Leiden, 1740 [translator was J. Stammetz with an introduction by W. la Bordus; reprinted in 1758 and 1772].

[95] Wolff (Christian) [1745] Kort Begrip der Grond-Beginzelen van alle de Mathematische Weetenschappen, Amsterdam, 1745 [two volumes; translated by A.F. Marci].