The ‘Chemistry of Space’: The Sources of Hermann Grassmann's Scientific Achievements

Abstract

Albert Lewis's article (Annals of Science, 1977) analysing the influence of Friedrich Schleiermacher on Hermann Grassmann, stimulated many different studies on the founder of n-dimensional outer algebra.

Following a brief outline of the various, sometimes diverging, analyses of Grassmann's creative thinking, new research is presented which confirms Lewis's original contribution and widens it considerably. It will be shown that:

i. Grassmann, although a self-taught mathematician, was at the centre of a hitherto understated intellectual trend, which was defining for Germany. Initiated by Pestalozzi's concept of elementary mathematical education and culminating in the modern mathematics of the late 19th Century, it was reflected in the contributions of Grassmann, Riemann, Jacobi and Eisenstein.

ii. Hermann Grassmann, his father Justus, and his brother Robert were all demonstrably influenced by Schleiermacher's dialectic; however the two brothers responded to it in very different ways.

iii. Whilst the more philosophical parts of Hermann's 1844 Extension Theory are characterised by the influence of Schleiermacher and also by the mathematical knowledge of his father, the entire development of this work is the unfolding of a single idea based on the father's interpretation of combinatorial multiplication as a ‘chemical conjunction‘, which was developed largely dialectically by Hermann.

Acknowledgements

I am deeply grateful to Albert C. Lewis for his careful revision of early versions, for all the very helpful remarks and suggestions incorporated in this paper.

I would also like to thank the anonymous referees for their helpful criticism. Without the organizational support and the helpful suggestions of Steve Russ the paper would not have been finished.

I dedicate this paper to my wife, Dr Karin Petsche.

Notes

¹ Hereafter the 1844 Ausdehnungslehre is abbreviated to A1: Hermann Grassmann, Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue mathematische Disciplin. Erster Theil: Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik, dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert. (Leipzig: Wiegand, 1844. 2. Ed. 1878) [Reprinted in Hermann Grassmann, Gesammelte mathematische und physikalische Werke. Vol. 1.1. (Leipzig: Teubner, 1894), 4–312]. [English translation by Lloyd Kannenberg in Hermann Grassmann, A New Branch of Mathematics (Chicago: Open Court, 1995), 3–312].The 1862 edition of the ‘Ausdehnungslehre’ is abbreviated to A2: Hermann Grassmann, Die Ausdehnungslehre. Vollständig und in strenger Form begründet (Berlin: Enslin, 1862). [Reprinted in Hermann Grassmann, Gesammelte mathematische und physikalische Werke. Vol. 1.2. (Leipzig: Teubner, 1896), 1–383]. [English translation by Lloyd Kannenberg, in Hermann Grassmann, Extension Theory. (AMS, 2000), 1–383].

² Edmund Husserl, ‘Studien zur Arithmetik und Geometrie. Texte aus dem Nachlass (1886–1901)’, Husserliana, Vol. xxi. (Boston: Kluwer, 1983), xxx.

³ A2 (note 1), 4.

⁴ Letter from A. F. Möbius to E. F. Apelt, 5 January 1846. Quoted from Friedrich Engel, Grassmanns Leben (Leipzig: Teubner, 1911) [= Grassmann, H. Gesammelte mathematische und physikalische Werke. Vol. 3.2.], 101.

⁵ Letter from J. A. Grunert to H. Grassmann, 9 December 1844. Quoted from Engel, 1911 (note 4), 103.

⁶ Letter from E. F. Apelt to A. F. Möbius, 3 September 1845. Quoted from Engel, 1911 (note 4), 101.

⁷ Moritz Cantor, ‘Grassmann, Hermann’, in Allgemeine Deutsche Biographie. Vol. 9 (Leipzig: Duncker & Humblot, 1879), 595–598 (596).

⁸ Robert Perseval Graves, Life of Sir William Rowan Hamilton. Vol. 3 (Dublin: Hodges & Figgis, 1889), 441–442.

⁹ Abbe, who studied under Weber (1804–1891) and Riemann (1826–1866), was a German mathematician and physicist, entrepreneur and social reformer. He held a chair at Jena University. Together with the mechanical scientist Carl Zeiss (1816–1888) and the chemist Otto Schott (1851–1935) he created in the Carl Zeiss company the foundations of modern optics and founded the world-famous Carl Zeiss brand.

¹⁰ Ernst Abbe, Briefe an seine Jugend- und Studienfreunde Carl Martin und Harald Schütz, edited by Volker Wahl and Joachim Wittig (Berlin: Akademie-Verlag, 1986), 159.

¹¹ Ernst Abbe, Briefe an seine Jugend- und Studienfreunde Carl Martin und Harald Schütz, edited by Volker Wahl and Joachim Wittig (Berlin: Akademie-Verlag, 1986), 214–215.

¹² The demands of his profession as a teacher forced him, he wrote to Hankel, to turn away from his efforts in the field of mathematics, ‘to give it up permanently, due to lack of suitable stimulus, with not even the fruits of intellectual community to be had, and experiencing only a wearing, destructive isolation…’ Letter from H. Grassmann to H. Hankel, 8. 12. 1866. In Engel, 1911 (note 4), 272.

¹³ H. Hankel in a letter to H. Grassmann in November 1866. In Engel, 1911 (note 4), 270. Hankel, a student of Riemann, was a friend of Ernst Abbe who met with him and received transcripts of Riemann's lectures from him, and who may likewise have learned of Grassmann's work from Abbe.

¹⁴ With reference to an initial overview of the reception of Grassmann by the above-mentioned mathematicians, philosophers and psychologists, I refer the reader to Hans-Joachim Petsche, Albert C. Lewis, Jörg Liesen and Steve Russ (eds.), From Past to Future: Grassmann's Work in Context. The Grassmann Bicentennial Conference, September 2009 (Basel: Birkhäuser, 2011).

¹⁵ Cf. Gert Schubring, ‘Remarks on the fate of Grassmann's Nachlaß’, in Schubring, 1996 (note 22), 19–25; Hans-Joachim Petsche, Hermann Grassmann. Biography. (Basel: Birkhäuser, 2009).

¹⁶ These documents may be found in Hans-Joachim Petsche, Lloyd Kannenberg, Gottfried Keßler and Jolanta Liskowacka (eds.), Hermann Grassmann – roots and traces. Autographs and unknown documents (Basel: Birkhäuser, 2009). They refer to: Justus Grassmann (1779–1852), the father (5 documents), Friedrich Heinrich Gotthilf Grassmann (1784–1866), the uncle (5 documents), Robert Grassmann (1815–1901), the brother (4 documents) and Hermann Günther Grassmann (1809–1877), (24 documents). In addition, in Petsche et al. 2011 (note 14) there are a further six documents on Hermann Grassmann as well as a self-published curriculum vitae of his brother Robert Grassmann.

¹⁷ A1 (note 1), 8, 26.

¹⁸ Albert C. Lewis, ‘H. Grassmann's 1844 Ausdehnungslehre and Schleiermacher's Dialektik, Annals of Science. 34 (1977), 103–162.

¹⁹ Hans-Joachim Petsche, Leben und Wirken Hermann Günther Grassmanns. 2 vol., (unpublished doctoral thesis, Pädagogische Hochschule Potsdam, 1979). (251 pp.).

²⁰ With the exception of the contribution of Karl-Heinz Schlote, ‘Hermann Grassmanns Beitrag zur Algebrentheorie’, Janus 72/4 (1985), 225–255.

²¹ These influences have reached even modern Schleiermacher research. See for example Andreas Arndt, ‘Einleitung’, in Friedrich Daniel Ernst Schleiermacher: Dialektik (1811), edited by Andreas Arndt (Hamburg: Felix Meiner, 1986), ix–lxxvi (xlv–xlvi).

²² Gert Schubring (ed.), Hermann Günther Grassmann (1809–1877): Visionary mathematician, scientist and neohumanist scholar. Papers from a sesquicentennial conference (Dordrecht, Boston and London: Kluwer, 1996).

²³ Cf. Gert Schubring, ‘The cooperation between Hermann and Robert Grassmann on the foundations of mathematics’, in Schubring, 1996 (note 22), 59–70.

²⁴ See Federigo Enriques, ‘Prinzipien der Geometrie’, in Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen. Vol. 3.1, Art. IIIA, Vol 1., commissioned by the Akademien der Wissenschaften zu Göttingen, Leipzig, München und Wien. (Leipzig: BG Teubner, 1907), 1–129 (63–64). Hermann Weyl remarks: ‘In arriving at a concept of a more than three-dimensional diversity, Grassmann as well as Riemann are influenced by the philosophical ideas of Herbart’. Hermann Weyl, Raum – Zeit – Materie. (1918). Vorlesungen über die allgemeine Relativitätstheorie, third edition (Berlin: Springer, 1919), 289. In more recent times Jammer writes: ‘Certain ideas in Herbart's philosophy seem to have had a great influence on Riemann and H. Grassmann in their formulation of a manifold with an arbitrary number of dimensions’. Max Jammer, Concepts of space. The history of theories of space in physics (New York: Dover, 1993), 177. See also Erik C. Banks, ‘Kant, Herbart and Riemann’, in Kant-Studien 96 (2) (2005), 208–234 (208).

²⁵ See Schubring, 1996 (note 23); id., ‘Hermann Günther Grassmann (1809–1877). Ein vielseitiger Innovator’, Mitteilungen der Deutschen Mathematiker-Vereinigung 17 (2009), 177–185; id., Gert Schubring, ‘Hermann Grassmann – zwei sich unterscheidende Lebensläufe’, NTM (N.S.) 18 (2010), 197–230.

²⁶ Cf. Schubring, 1996 (note 23); Dominique Flament, ‘Théorie des formes et avènement d'une nouvelle discipline des mathématiques pures, selon Hermann Günther GRASSMANN (1809–1877)’, Revista Brasileira de História da Matemática, 1/2, (July/December, 2008), 178–210.

²⁷ Albert C. Lewis, ‘The unity of logic, pedagogy and foundations in Grassmann's mathematical work’, History and Philosophy of Logic 25 (2004), 15–36.

²⁸ So Schubring, 2009 (note 25) and 2010 (note 25).

²⁹ Cf. Albert C. Lewis, ‘Justus Grassmann's school programs as antecedents of Hermann Grassmann's 1844 “Ausdehnungslehre”’, in Hans Niels Jahnke and Michael Otte (eds.), Epistemological and Social Problems of the Sciences in the Early 19th Century (Dordrecht: Reidel, 1981), 255–268; Erhard Scholz, ‘The influence of Justus Grassmann's crystallographic works on Hermann Grassmann’, in Schubring, 1996 (note 22), 37–45; Mircea Radu, ‘Justus Grassmann's contributions to the foundations of mathematics – mathematical and philosophical aspects’, Historia Mathematica 27 (2000), 4–35.

³⁰ Cf. Marie-Luise Heuser, ‘Geometrical product – exponentiation – evolution. Justus Günther Grassmann and dynamist Naturphilosophie’, in Schubring, 1996 (note 22), 47–58.

³¹ Cf. Michael Otte, ‘The ideas of Hermann Grassmann in the context of the mathematical and philosophical tradition since Leibniz’, Historia Mathematica 16 (1989), 1–35.

³² See Hans Niels Jahnke, Mathematik und Bildung in der Humboldtschen Reform (Goettingen: Vandenhoeck + Ruprecht, 1990); Radu, 2000 (note 29).

³³ Further information on the Stettin schoolmaster Friedrich Heinrich Gotthilf Grassmann and the foster father Robert Grassmann can be found in Petsche et al., 2009 (note 16), 59–87.

³⁴ Cf. Engel, 1911 (note 4), 132–134, 155, 225.

³⁵ Cf. Schubring, 1996 (note 23); Radu, 2000 (note 29); Hans-Joachim Petsche, Grassmann (Basel: Birkhäuser, 2006); Ivor Grattan-Guinness, ‘Discovering Robert Grassmann (1815–1901)’, in Petsche et al., 2011 (note 14), 19–35.

³⁶ See Petsche et al., 2011 (note 14).

³⁷ Lewis, 1977 (note 18); Petsche, 1979 (note 19).

³⁸ Johann Heinrich Pestalozzi, ‘Wie Gertrud ihre Kinder lehrte. Ein Versuch, den Müttern Anleitung zu geben, ihre Kinder selbst zu unterrichten; in Briefen (1801)’, in Johann Heinrich Pestalozzi, Werke, Vol. 5 (Stuttgart und Tübingen: Cotta'sche Buchhandlung 1820). Roger Vaissière stresses in this context: ‘Herbart clearly had a formative experience with Pestalozzi’. Roger Vaissière, Pestalozzi als Pädagoge und Bildungspolitiker. Referat am Pestalozzi-Symposium, 10. April 2008, Langenthal <http://www.heinrich-pestalozzi.de/de/dokumentation/veranstaltungen/pestalozzis_langenthaler_rede/pestalozzi_als_paedagoge_und_bildungspolitiker/index.htm#print> [accessed 21 July 2013].

³⁹ Johann Friedrich Herbart, ‘Über Pestalozzi's neueste Schrift: Wie Gertrud ihre Kinder lehrte. An drei Frauen. [1802]’. in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 137–150 (140--141).

⁴⁰ See Heinrich Philipp Sextro (1746–1838), Ueber die Bildung der Jugend zur Industrie. Ein Fragment. (Göttingen: Dieterich, 1785).

⁴¹ Johann Heinrich Pestalozzi, ‘Über Volksbildung und Industrie’, in Johann Heinrich Pestalozzi. Über Volksbildung und Industrie. Zweck und Plan einer Armenerziehungsanstalt (1806/7), edited by Heinz Mühlmeyer (Heidelberg: Quelle & Meyer, 1964), 5–28 (8).

⁴² Pestalozzi 1801 (note 38), 114.

⁴³ Pestalozzi 1801 (note 38), 117. While number and form in the realm of elementary mathematical education are the focus of considerations in the following, language had its own independent place in elementary language education. It may suffice here to mention that the brothers F.H. Grassmann and J. Grassmann (Hermann's father) divided the implementation of the Pestalozzian ideas between them: J Grassmann undertook mathematics and F.H. Grassmann the study of language. See: Justus Günther Grassmann, Raumlehre für Volksschulen, 1. Teil: Ebene räumliche Verbindungslehre (Berlin: Realschulbuchhandlung, 1817); Raumlehre für die untern Klassen der Gymnasien, und für Volksschule, 2. Teil: Ebene räumliche Größenlehre (Berlin: Reimer, 1824); Friedrich Heinrich Grassmann, Sprachbildungslehre für das Deutsche, Teil 1: Die Lehre von der Sylbenbildung (Berlin: Reimer, 1828); Teil 2: Die Lehre von deutscher Wortbildung (Berlin: Reimer, 1829); Teil 3: Die Lehre von Deutscher Redebildung (Berlin: Reimer, 1830). We may also remark that in Part I of his theory of language learning, F.H. Grassmann, entirely in keeping with the meaning of the elementary method, brought into play combinatorics and phonetics in order, among other things, to determine the set of possible syllables for the German language.

⁴⁴ Johann Heinrich Pestalozzi, ‘Bericht an die Eltern und an das Publikum über den gegenwärtigen Zustand und die Einrichtung der Pestalozzischen Anstalt in Iverten, Februar 1808’, in Johann Heinrich Pestalozzi, Sämtliche Werke. Kritische Ausgabe Bd. 21. Schriften von Ende 1808 bis Anfang 1809 (Zürich: Orell Füssli, 1958), 11–88 (76).

⁴⁵ Johann Heinrich Pestalozzi, ‘Elementarbildung zur Industrie. Um 1807–1808’, in Johann Heinrich Pestalozzi, Sämtliche Werke. Kritische Ausgabe. Bd. 20. Schriften von Ende 1806 bis Anfang 1808 (Zürich: Orell Füssli, 1956), 295–304 (299).

⁴⁶ Pestalozzi, 1806/7 (note 41), 12.

⁴⁷ See Johann Beckmann, ‘Entwurf der algemeinen Technologie’, in Vorrath kleiner Anmerkungen über mancherley gelehrte Gegenstände. Third Part, edited by Johann Beckmann (Göttingen: Röwer, 1806), 463– 533. For more detail on the significance of Beckmann see Gerhard Banse (ed.), Allgemeine Technologie zwischen Aufkärung und Metatheorie. Johann Beckmann und die Folgen. (Berlin: edition sigma, 1997).

⁴⁸ Pestalozzi, 1801 (note 38), 80.

⁴⁹ Adolf Diesterweg, ‘Selbstbesprechung der “Raumlehre oder Geometrie”, Bonn 1843’, in Adolf Diesterweg (ed.), Wegweiser zur Bildung für deutsche Lehrer. Vol 2, 4th rev. ed. (Essen: Bädeker, 1851), 424–428 (424–425).

⁵⁰ Adolf Diesterweg, ‘Selbstbesprechung der “Raumlehre oder Geometrie”, Bonn 1843’, in Adolf Diesterweg (ed.), Wegweiser zur Bildung für deutsche Lehrer. Vol 2, 4th rev. ed. (Essen: Bädeker, 1851), 426.

⁵¹ Birgit Ofenbach, Geschichte des pädagogischen Berufsethos. Realbedingungen für Lehrerhandeln von der Antike bis zum 21. Jahrhundert (Würzburg: Königshausen & Neumann, 2006), 127.

⁵² Ofenbach, 2006 (note 51), 113. Likewise Pestalozzi himself remarked: ‘The Prussian state is the first and only one that, through persistent encouragements, not only acted charitably towards my undertaking, but even sought to make use of it, which was something only wished for up until then.’ Johann Heinrich Pestalozzi, Sämtliche Briefe. Bd. 10: Briefe aus den Jahren 1816 und 1817 (Zürich: Orell Füssli, 1968), 411.

⁵³ On 30 October even the solid fortress of Stettin surrendered to the French without a fight when faced with ‘a small band of 800 enemy horsemen’. Martin Wehrmann, Geschichte der Stadt Stettin (Stettin: Leon Sauniers Buchhandlung, 1911), 417. See also figure 2.

⁵⁴ Letter from Schleiermacher to Charlotte von Kathen, 20.6.1806. In Heinrich Meisner (ed.), Schleiermacher als Mensch. Sein Werden und Wirken. 2. Teil: Sein Wirken. Familien- und Freundesbriefe 1804–1834. (Gotha: Perthes, 1923), 64.

⁵⁵ Letter from Schleiermacher to Henriette von Herz, December 1806. In Meisner, 1923 (note 54), 81.

⁵⁶ Johann Gottlieb Fichte, Addresses to the German nation (1808), translated by Isaac Nakhimovsky, Bêla Kapossy, and Keith Tribe (Indianapolis: Hackett Publishing Company, 2013), 109.

⁵⁷ Johann Gottlieb Fichte, Addresses to the German nation (1808), translated by Isaac Nakhimovsky, Bêla Kapossy, and Keith Tribe (Indianapolis: Hackett Publishing Company, 2013), 114.

⁵⁸ Johann Gottlieb Fichte, Addresses to the German nation (1808), translated by Isaac Nakhimovsky, Bêla Kapossy, and Keith Tribe (Indianapolis: Hackett Publishing Company, 2013), 119. Hence elementary mathematical education for Fichte almost takes on a greater (socio-political) significance than mathematical education at academic secondary schools and universities. Cf. Jahnke, 1990 (note 32).

⁵⁹ Friedrich Schleiermacher, Occasional Thoughts on Universities in the German Sense, with an appendix regarding a university soon to be established (1808), translated and annotated by Terrence N. Tice, with Edwina Lawler (San Francisco: EM Text., 1991), 52.

⁶⁰ Friedrich Schleiermacher, Occasional Thoughts on Universities in the German Sense, with an appendix regarding a university soon to be established (1808), translated and annotated by Terrence N. Tice, with Edwina Lawler (San Francisco: EM Text., 1991), 51.

⁶¹ Letter from Schleiermacher to Karl Gustav von Brinckmann, September 1811. In Meisner, 1923 (note 54), 138.

⁶² Cf. Ingrid Lohmann, ‘Über den Beginn der Etablierung allgemeiner Bildung. Friedrich Schleiermacher als Direktor der Berliner wissenschaftlichen Deputation’, Zeitschrift für Pädagogik 30/6 (1984), 749–773; Petsche, 2006 (note 35), 272–273.

⁶³ See in particular the report written by Bartholdy for the Berlin Scientific Deputation, written in 1810: ‘On the stages, methods and extent of mathematical and scientific teaching’, discussed in Lohmann, 1984 (note 62).

⁶⁴ ‘One is generally doing the working classes an injustice…’, Schleiermacher wrote in 1813 to Sophie Marie Gräfin von Voß (1729–1814), ‘…if one credits them only with strength and considers them coarse. … I can see equally clearly that from below everything looks so nice, and is just as we would wish it to be, and the main issue remains this: how much can be ruined from above?’. Letter from Schleiermacher to Countess Sophie Marie von Voss, 07.6.1813. In Meisner, 1923 (note 54), 183. For Bartholdy's activity in the spirit of Pestalozzi, see later references.

⁶⁵ Quoted from Helmut König, Zur Geschichte der bürgerlichen Nationalerziehung in Deutschland zwischen 1807 und 1815. Teil 1. (Berlin: Volk und Wissen, 1972), 323.

⁶⁶ Jahnke, 1990 (note 32), 1.

⁶⁷ For the state of the elementary and primary school system in Germany, see for example Johann Wilhelm David Korth (ed.), Ökonomisch-technologische Encyklopädie, oder allgemeines System der Staats-, Stadt-, Haus- und Landwirthschaft, und der Kunstgeschichte in alphabetischer Ordnung. Part 149, Schuld bis Schalbacher Brunnen (Berlin: Paulinische Buchhandlung, 1828), keyword 'Schulstube’, 542–553; Konrad Fischer, Geschichte des Deutschen Volksschullehrerstandes. Vol 1. Von dem Ursprunge der Volksschule bis 1790 (Hannover: Meyer, 1892); Wolfgang Neugebauer, Absolutistischer Staat und Schulwirklichkeit in Brandenburg-Preussen (Berlin and New York: de Gruyter, 1985; id. (ed.), Schule und Absolutismus in Preußen. Akten zum preußischen Elementarschulwesen bis 1806 (Berlin and New York: de Gruyter, 1992), 548–552.

⁶⁸ Friedrich Heinrich Grassmann: Urkunde der Gründung einer Stiftung zur Erhöhung des Einkommens sehr gering dotirter Landschulen in Pommern (Document for Creating a Fund to Raise the Income of Poorly-Funded Country Schools in Pomerania) (21st May 1866). In Petsche et al., 2009 (note 16), 86.

⁶⁹ Quoted from Wübbe Ulrichs Jütting, Geschichte des Rückschritts in der Dotation der preußischen Volksschule. Beiträge zur innern Schulgeschichte und zur Kritik der bestehenden Schulgesetzgebung nebst der Unterrichtsgesetzes-Vorlage (Minden: Aug. Volkening, 1870), 6.

⁷⁰ Cf. Karl Justus Blochmann, Heinrich Pestalozzi. Züge aus dem Bilde seines Lebens und Wirkens nach Selbstzeugnissen, Anschauungen und Mittheilungen (Leipzig: Brockhaus, 1846), 157–158; Eduard Clausnitzer, ‘Die Volksschullehrerbildung’, in Lexis, W. (ed.), Das Volksschulwesen und das Lehrerbildungswesen im Deutschen Reich. (Berlin: Asher & Co, 1904), 233– 341 (240).

⁷¹ Blochmann, 1864 (note 70), 158.

⁷² See Karl Scheibert, Geschichte des Geschlechts Grassmann und seiner Nebenlinien (Görlitz: Starke, 1937), 36.

⁷³ Thus as early as 1803, von Gneisenau (1760–1831) delivered a paper to the Prussian king, which for the most part was enthusiastic about the efficiency of Pestalozzi's method as a means of raising the level of elementary schools. See Heinz-Elmar Tenorth, Geschichte der Erziehung. Einführung in die Grundzüge ihrer neuzeitlichen Entwicklung (Weinheim and München: Juventa, 2008), 154–155.

⁷⁴ See Barbara Vinken, Die deutsche Mutter. Der lange Schatten eines Mythos (München: Piper2001); Sabine Kebir, ‘Die deutsche Mutter’, Ossietzky 2 (2002), <http://www.sopos.org/aufsaetze/3c7655a074a52/1.phtml> [accessed 21 July 2013]; Aimee Waesche and Nina Slawik. Der Sonderweg der deutschen Mutter (06.09.2006) <http://www.familienheute.de/attachments/120_Der Sonderweg der deutschen Mutter.pdf> [accessed 21 July 2013]. A ‘mother-centric’ interpretation of Pestalozzi still persists even today. See also Theodor Ballauff and Klaus Schaller, Pädagogik. Eine Geschichte der Bildung und Erziehung. Vol. 2: Vom 16. bis zum 19. Jahrhundert. (Freiburg/München: Alber, 1970), 480–481.

⁷⁵ Friedrich Heinrich Grassmann had the same view of this problem, namely that youth would not come to adhere to social norms if they were conditioned too early in life to go around asking ‘why?’. This is how ‘communists and atheists would develop, who would later […] help to create a situation such as we witnessed in 1848’. Friedrich Heinrich Gottlieb Grassmann, ‘Bruchstücke über den grundlegenden Unterricht, besonders für die Sprache (den sprachlichen Elementarunterricht)’, Erste Abtheilung der Pädagogischen Revue. Section 1, 36 (1854), 81–100 (97). Independent thought should therefore be encouraged in keeping with the level of maturity of the child.

⁷⁶ ‘Diesterweg and his hangers-on’ had ruined primary school education with their ‘excessive demands’, says the Real-Encyklopädie des Erziehungs- und Unterrichtswesens of 1866. ‘The 1848 movement exposed what damage had been done to schools in an extremely lamentable manner, and we realised to our horror that we were reaping a whirlwind after sowing a breeze’. Hermann Rolfus and Adolph Pfister, Real-Encyclopädie des Erziehungs- und Unterrichtswesens nach katholoischen Prinicipien. Vol 4. (Mainz: Florian Kupferberg, 1866), 9–10.

⁷⁷ See Moritz Cantor, ‘Ferdinand Schweins und Otto Hesse’, in Heidelberger Professoren aus dem 19. Jahrhundert: Festschrift zur Zentenarfeier ihrer Erneuerung durch Karl Friedrich. – Vol 2, (Heidelberg: Winter, 1903), 221–242 (224).

⁷⁸ Fischer, who in this context became involved in lengthy and depressing accusations of plagiarism, was himself a close friend of Justus Grassmann. See also footnote 95.

⁷⁹ Carl Friedrich Hindenburg, ‘Die Combinationslehre ist eine selbständige Grundwissenschaft’, in Carl Friedrich Hindenburg (ed.), Sammlung combinatorisch-analytischer Abhandlungen. Erste Sammlung (Leipzig: Fleischer d.J. 1796), 153–304 (302).

⁸⁰ Carl Friedrich Hindenburg, ‘Die Combinationslehre ist eine selbständige Grundwissenschaft’, in Carl Friedrich Hindenburg (ed.), Sammlung combinatorisch-analytischer Abhandlungen. Erste Sammlung (Leipzig: Fleischer d.J. 1796), 153.

⁸¹ Hans Niels Jahnke praised unreservedly the significance of mathematics for education and its interlinking with philosophy and emphasised the significance of the Hindenburgian Combinatorial School at the beginning of the 19^th Century. See Jahnke, 1990 (note 32); Hans Niels Jahnke, ‘Mathematik und Romantik. Die Aphorismen des Novalis zur Mathematik’, UNIKATE 33 (2008), 30–41, and also Philippe Séguin, ‘Von der Philosophie zur ars combinatoria. Novalis' Erwartungen an die Mathematik und die Folgen’, in Andrea Albrecht, Gesa von von Essen and Werner Frick, Zahlen, Zeichen und Figuren: mathematische Inspirationen in Kunst und Literatur (Berlin and Boston: de Gruyter, 2011), 248–267.

⁸² Cf. Jahnke, 1990 (note 32), 171–172.

⁸³ 'Topics' here refers to a method of presentation and discovery of truths situated between logic, dialectic, and rhetoric, and which is tied to the names of Raimundus Lullus (1232–1316), Georgius Agricola (1494–1555), Athanasius Kircher (1602–1680), and Gottfried Wilhelm Leibniz. Since Leibniz, the increasingly better known (mathematical) combinatorics served this purpose. See e.g. Wilhelm Schmidt-Biggemann, Topica Universalis: Eine Modellgeschichte humanistischer und barocker Wissenschaft (Hamburg: Meiner, 1983).

⁸⁴ Cantor, 1903 (note 77), 224–225.

⁸⁵ Johann Friedrich Lorenz, Lehrbegriff der Syntactik, oder Combinationslehre (Magdeburg: Georg Christian Keil, 1806), 20.

⁸⁶ Johann Friedrich Lorenz, Lehrbegriff der Syntactik, oder Combinationslehre (Magdeburg: Georg Christian Keil, 1806), 20.

⁸⁷ See Johann Friedrich Lorenz, Lehrbegriff der Syntactik, oder Combinationslehre (Magdeburg: Georg Christian Keil, 1806), 20.

⁸⁸ See Jakob Friedrich Fries, Die mathematische Naturphilosophie nach philosophischer Methode bearbeitet (Heidelberg: Winter, 1822), 63–65. For more detail on this, see Hans-Joachim Petsche, ‘Schleiermacher, Fries, Herbart … – wer beeinflusste Hermann Grassmann?’, Mathematische Semesterberichte, 59/2 (2012), 183–222.

⁸⁹ Paul Bernays, The Basic Ideas of Friesean Philosophy in Relation to Contemporary Science (1930), translation by Volker Peckhaus (Bernays Project, 2004) <http://www.learningace.com/doc/2103964/6f56d0d90d7eec72376159bee4a3edba/bernays10_2004-02-15> [accessed 21 July 2013], 11.

⁹⁰ See Karl Christian Friedrich Krause, ‘Rezension zu ‘Lorenz’ Lehrbegriff der Syllogistik’’, Neue Leipziger Literatur-Zeitung. Stück 121, (1807), 2097–2105.

⁹¹ Krause 1807 (note 90), 2099. It is noteworthy that Pestalozzi's pupil Friedrich Fröbel (1782–1852) radicalised the dynamics of construction in the proto-mathematical field of the combinatorics of children's play with spatial elements. See Yasuhiro Shouji, 2001. ‘Die Kunde der Formen und Gestalten der Spielgaben. Fröbel und sein ‘Körperkasten’’, in Helmut Heiland, Elisabeth Gutjahr and Karl Neumann (eds.), Fröbel-Forschung in der Diskussion. Internationale Ergebnisse zu methodologischen und rezeptionsgeschichtlichen Fragen (Weinheim: Beltz, 2001), 88–101 (99–100). On account of this Fröbel earned a positive mention by Felix Klein. Cf. Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Teil 1, edited by Richard Courant and Otto Neugebauer (Berlin: Springer, 1926), 128. Fröbel also showed great spiritual affinity with his friend Krause and there is evidence that he came under Schleiermacher's influence. See Alexander Bruno Hanschmann, Friedrich Fröbel. Die Entwickelung seiner Erziehungsidee in seinem Leben. Nach authentischen Quellen dargestellt (Eisenach: Bacmeister, 1874), 90, 94, 148–150, 153–154.

⁹² See Justus Günther Grassmann, Zur physischen Krystallonomie und geometrischen Combinationslehre, Erstes Heft (Stettin: Morin, 1829) 175.

⁹³ See Christian August Semler, ‘Ueber die combinatorische Methode’, Neues allgemeines Intelligenzblatt für Literatur und Kunst. Zur Neuen Leipziger Literatur-Zeitung gehörend. 1. Stück. Januar 1809, 3–12 (4); id., Versuch ueber die combinatorische Methode, ein Beytrag zur angewandten Logik und allgemeinen Methodik, 2nd edn. (1st edn., 1811) (Dresden: Walthersche Hofbuchhandlung, 1822). See also footnote 83.

⁹⁴ See Josef Leonhard Blass, Herbarts pädagogische Denkform oder Allgemeine Pädagogik und Topik. (Ratingen: Henn, 1969).

⁹⁵ Diesterweg, who places combinatorial thinking increasingly at the centre of the mathematical curriculum, quoted him as follows: ‘The celebrated writer on teaching and scientific matters, E.G. Fischer in Berlin, says in one of his works: ‘Combinatorics is by its very nature not confined to mathematics. Because all imaginable things can be combined, whether directly or via symbols. Hence, audible language is a combination of sounds, and all judgement is a combination of imagination and concepts; all thought is a combining of judgements. So indeed it can be said that all our mental activity consists in a kind of combining.’’. Adolf Diesterweg, ‘Die Wichtigkeit des Combinirens im Unterrichte’, Rheinische Blätter für Erziehung und Unterricht mit besonderer Berücksichtigung des Volksschulwesens, NF, Vol 3, H. 2 (1831), 193–212, 210.This is the same Fischer referred to by Justus Grassmann in his Crystallonomy (J. Grassmann, 1829 (note 92), 10), and in his Textbook of Geometry, as his ‘esteemed friend’, and whose treatment of the Euclidean parallel axioms he was entirely in agreement with! See J. Grassmann, 1824 (note 43), xxiv. This proximity to Fischer is also highlighted by the fact that Fischer too was an admirer of Pestalozzi's methods. See Ernst Gottfried Fischer, ‘Ueber Pestalozzi's Lehrart’, in Sammlung der deutschen Abhandlungen, welche in der Königlichen Akademie der Wissenschaften zu Berlin vorgelesen worden in dem Jahre 1803 (Berlin: Georg Decker, 1806), 1181–1190. Fischer took a similar stance on the topics of dynamism, atomism and speculation on nature (Naturspekulation) to J. Grassmann. Cf. Ernst Gottfried Fischer, Lehrbuch der mechanischen Naturlehre (Berlin: G.C. Nauck, 1805), 18–29.

⁹⁶ Quoted in [Herbart, 1887, lxiii/lxiv].

⁹⁷ Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 151–274 (157).

⁹⁸ Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 174.

⁹⁹ Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 174.

¹⁰⁰ Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 163.

¹⁰¹ Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 163.

¹⁰² See Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 176.

¹⁰³ See Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 180.

¹⁰⁴ See Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 178.

¹⁰⁵ See Cf. Johann Friedrich Herbart, ‘Pestalozzi's Idee eines ABC der Anschauung. 1802 und 1804’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Vol. 1 (Langensalza: Hermann Beyer & Söhne, 1887), 179–180.

¹⁰⁶ Pestalozzi's main element of form was the rectangle, which was easier to use with pupils than the triangle. Herbart was also, in contrast with Pestalozzi, mostly interested in the gifted pupils who were destined for scientific studies. While Pestalozzi dealt with peasant children, Herbart was concernd with developing a scientific pedagogy for private tutors. See Ofenbach, 2006 (note 51), 143–148. Hence Pestalozzi's subsequent judgement of Herbart, as made to Torlitz: ‘Herbart's triangle is the rectangle of the upper classes’. Quoted from: Remark on ,Antwort of neun Fragen Herbarts über die Methode’, 2. Appendix. In Johann Heinrich Pestalozzi, Sämtliche Werke. Kritische Ausgabe Bd. 15. Schriften von Ende 1803-1804 (Zürich: Orell Füssli, 1958), 539.

¹⁰⁷ On the problem of teaching trigonometry, he wrote: ‘It is perhaps particularly useful, in order to achieve the advantage mentioned in the introduction, to present not only individual magnitudes, but also the great mass of triangles, as fluid, i.e. caught up in permanent transitions. Even the meaning of differential formulae in trigonometry could thus be made intuitively clear in advance’. Herbart, 1802/04 (note 97), 186–187.

¹⁰⁸ It is interesting to note that modern computers utilise this same principle of triangulation in parallel processing.

¹⁰⁹ Johann Friedrich Herbart, ‘Allgemeine Pädagogik aus dem Zweck der Erziehung abgeleitet. [1806.]’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Bd. 2 (Langensalza: Hermann Beyer & Söhne, 1887), 1–139 (57).

¹¹⁰ Johann Friedrich Herbart, ‘Allgemeine Pädagogik aus dem Zweck der Erziehung abgeleitet. [1806.]’, in Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Bd. 2 (Langensalza: Hermann Beyer & Söhne, 1887), 61.

¹¹¹ On mineralogy: ‘It is very difficult indeed to think of a better opportunity to school the eye to pick up the smallest variations in texture, sheen and colour, and thereby to combine this with so many other sensory perceptions’. Herbart, 1802/04 (note 97), 247.

¹¹² Herbart, 1806 (note 109), 74–76.

¹¹³ Dealing with Kant's synthetic judgement, he writes: ‘In order to discuss this subject in the most general manner possible, I would first of all wish to draw an example from combinatorics, which logically speaking ranks higher than all mathematics’. Johann Friedrich Herbart, Psychologie als Wissenschaft. Neu gegründet auf Erfahrung, Metaphysik und Mathematik. Zweiter, analytischer Teil. 1825. In Johann Friedrich Herbart, Sämtliche Werke. In chronlogischer Reihenfolge, edited by Karl Kehrbach. Bd. 6 (Langensalza: Hermann Beyer & Söhne, 1892), 1–338 (238).

¹¹⁴ Blass, 1969 (note 94), 151.

¹¹⁵ Among Erhard Scholz's discoveries in Riemann's personal papers was an excerpt which ‘noted Herbart's conceptions of things as bundles [Complexionen] of properties (R.59), behind which lay the real’. See Erhard Scholz, ‘Herbart's Influence on Bernhard Riemann’, Historia Mathematica 9 (1982), 413-440, 419.

¹¹⁶ See Bernhard Riemann, ‘Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. (Habilitationsschrift, 1854, aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.)’, in Bernhard Riemann's gesammelte mathematische Werke und wissenschaftlicher Nachlaß, edited by Heinrich Weber and Richard Dedekind (Leipzig: B. G. Teubner, 1876), 254–269.

¹¹⁷ Erhard Scholz has shown that it is exactly those two points that influenced Riemann's conception of n-dimensional manifolds. See Scholz, 1982. Scholz also noted that Riemann ‘transformed certain features of Herbart's philosophy into guiding principles of mathematics which then operated in his own work. This was exactly what Herbart had expected philosophy could do in relation to science’. See Bernhard Riemann, ‘Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. (Habilitationsschrift, 1854, aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.)’, in Bernhard Riemann's gesammelte mathematische Werke und wissenschaftlicher Nachlaß, edited by Heinrich Weber and Richard Dedekind (Leipzig: B. G. Teubner, 1876), 428. In this context, see also Erhard Scholz, ‘Bernhard Riemanns Auseinandersetzung mit der Herbartschen Philosophie’, in Andreas Hoeschen and Lothar Schneider (eds.), Herbarts Kultursystem. Perspektiven der Transdisziplinarität im 19. Jahrhundert (Würzburg: Königshausen & Neumann, 2001), 163–184.

¹¹⁸ A1 (note 1), 27.

¹¹⁹ There was a lively exchange between Bartholdy and Schleiermacher on the subject of activities designed to promote elementary school education. Schleiermacher wrote to Gass about Bartholdy: ‘He told me something in Berlin about his plan for a training college; I liked this greatly, and rejoice in the fact that I had understood Pestalozzi's thinking in the same way as he had’. Letter from Schleiermacher to Gass, May 1805. In Friedrich Schleiermacher, Briefwechsel mit Joachim Christian Gaß, edited by Wilhelm Gaß (Berlin: Reimer, 1852), 23.

¹²⁰ Carl Scheibert, ‘Nekrolog’, Pädagogische Revue. Zweite Abteilung, 32 (June, 1852), 202–205, (203).

¹²¹ J. Grassmann, 1817 (note 43), iv.

¹²² ‘Räumliche Verbindungslehre’ (Spatial Theory of Conjunction) is for Grassmann the German term for ‘geometric combination theory’. See J. Grassmann, 1817 (note 43), xi–xii.

¹²³ Adolf Diesterweg, ‘Besprechung der Raumlehren Justus Grassmanns von 1817, 1824 und 1826’, Rheinische Blätter für Erziehung und Unterricht mit besonderer Berücksichtigung des Volksschulwesens. Vol 1, H. 2, (1827), 106–110 (108).

¹²⁴ See J. Grassmann, 1817 (note 43), vii, viii; id. 1824 (note 43), x.

¹²⁵ See Joseph Schmid, Die Elemente der Form und Größe (gewöhnlich Geometrie genannt) nach Pestalozzi's Grundsätzen bearbeitet. 1. und 2. Teil (Bern: Wittwe Stämpfli, 1809), xxii.

¹²⁶ See Joseph Schmid, Die Elemente der Form und Größe (gewöhnlich Geometrie genannt) nach Pestalozzi's Grundsätzen bearbeitet. 1. und 2. Teil (Bern: Wittwe Stämpfli, 1809), xxiii.

¹²⁷ See Joseph Schmid, Die Elemente der Form und Größe (gewöhnlich Geometrie genannt) nach Pestalozzi's Grundsätzen bearbeitet. 1. und 2. Teil (Bern: Wittwe Stämpfli, 1809), xxiv.

¹²⁸ J. Grassmann, 1824 (note 43), v.

¹²⁹ J. Grassmann, 1817 (note 43), viii.

¹³⁰ J. Grassmann, 1817 (note 43), x.

¹³¹ J. Grassmann, 1817 (note 43), xi.

¹³² J. Grassmann, 1817 (note 43), ix.

¹³³ J. Grassmann, 1817 (note 43), 18.

¹³⁴ J. Grassmann, 1817 (note 43), 57.

¹³⁵ J. Grassmann, 1817 (note 43), 1.

¹³⁶ Diesterweg, 1827 (note 123), 108. Justus Grassmann remarks about his method: ‘I would like to name this method the real ‘synthetic method’, but not in the same sense as the word is used in mathematics, especially as it has as one of its laws the combinatorial progression from the simple to the more composite’. J. Grassmann, 1824 (note 43), xi. ‘That which sets out to form must itself be formed’. J. Grassmann, 1817 (note 43), xv.

¹³⁷ Quoted from Lohmann, 1984 (note 62), 759.

¹³⁸ J. Grassmann, 1824 (note 43), v.

¹³⁹ Elementary mathematics teaching was also the driving force behind his work on crystallonomy. He recorded in the introduction to his Crystallonomy that he had ‘preceded geometric combination theory with a number of exercises in order to allow the pupils to orientate themselves in space […]; the intention was at the same time to arrive at a sound basis for spatial extension, direction and movement […]. This in turn led to the study of crystallography’. J. Grassmann, 1829 (note 92), x/xi.

¹⁴⁰ Hermann Grassmann refers directly to this in his Extension Theory of 1844. See A1 (note 1), 8.

¹⁴¹ Justus Günther Grassmann, Ueber den Begriff und Umfang der reinen Zahlenlehre. Programmabhandlung des Stettiner Gymnasiums (Stettin, 1827).

¹⁴² See J. Grassmann, 1829 (note 92).

¹⁴³ See J. Grassmann, 1829 (note 92)., ix.

¹⁴⁴ See Scholz, 1996 (note 29), 41.

¹⁴⁵ That Justus Grassmann did not happen upon the obvious idea of (vector) addition of displacements seems due to two factors, namely using the infinitely-long straight line as the starting-point in geometric combination theory and the understanding of addition as a logical synthesis. Hermann Grassmann developed vector addition in the early 1830's whilst Justus Grassmann developed the geometrical multiplication of displacements in 1824 – see J. Grassmann, 1824 (note 43), 194 –, but did not pursue this idea further.

¹⁴⁶ J. Grassmann, 1829 (note 92), 34.

¹⁴⁷ Cf. J. Grassmann, 1829 (note 92), 34–35.

¹⁴⁸ Cf. J. Grassmann, 1829 (note 92), 35.

¹⁴⁹ A1 (note 1), 27.

¹⁵⁰ More on this in section 4.1.

¹⁵¹ J. Grassmann, 1829 (note 92), 70.

¹⁵² It should be noted that Christian Weiss was the elder brother of mineralogist Christian Samuel Weiss, whose influence in the field of natural philosophy on Justus Grassmann is discussed by Marie-Luise Heuser, ‘The Significance of Naturphilosophie for Justus and Hermann Grassmann’, in Petsche et al., 2011 (note 14), 49–59.

¹⁵³ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 68–112; 221–265 (68).

¹⁵⁴ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 69.

¹⁵⁵ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 92

¹⁵⁶ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 225.

¹⁵⁷ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 229.

¹⁵⁸ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 232.

¹⁵⁹ See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 233.

¹⁶⁰ Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 235.

¹⁶¹ See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 236.

¹⁶² See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 240.

¹⁶³ See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 240–241

¹⁶⁴ See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 242.

¹⁶⁵ See Ernst Tillich, ‘Ausführliche Analyse von Pestalozzis Schrift; Wie Gertrud Ihre Kinder lehrt’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 1 (1803), 243.

¹⁶⁶ Ernst Tillich, ‘Wissenschaftliche Darstellung der arithmetischen und geometrischen Anschauung, mit Rücksicht auf den mathematischen …’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 2 (1805), 1–74; 127–168, (5–6).

¹⁶⁷ Ernst Tillich, ‘Wissenschaftliche Darstellung der arithmetischen und geometrischen Anschauung, mit Rücksicht auf den mathematischen …’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode 2 (1805), 1–74; 127–168, 5.

¹⁶⁸ Ernst Tillich, Allgemeines Lehrbuch der Arithmetik oder Anleitung zur Rechenkunst für Jedermann (Leipzig: Gräff, 1806), 407

¹⁶⁹ Ernst Tillich, Allgemeines Lehrbuch der Arithmetik oder Anleitung zur Rechenkunst für Jedermann (Leipzig: Gräff, 1806), 407–408.

¹⁷⁰ Ernst Tillich, Allgemeines Lehrbuch der Arithmetik oder Anleitung zur Rechenkunst für Jedermann (Leipzig: Gräff, 1806), 408.

¹⁷¹ Ernst Tillich, Allgemeines Lehrbuch der Arithmetik oder Anleitung zur Rechenkunst für Jedermann (Leipzig: Gräff, 1806), 408.

¹⁷² Tillich, 1805 (note 166), 29.

¹⁷³ Tillich, 1805 (note 166), 31.

¹⁷⁴ Tillich, 1805 (note 166), 36.

¹⁷⁵ Ernst Tillich, ‘Anzeige: Pestalozzis Idee eines ABC der Anschauung untersucht und wissenschaftlich ausgeführt von J. F. Herbart. Göttingen 1802’, Beiträge zur Erziehungskunst, zur Vervollkommnung sowohl ihrer Grundsätze als ihrer Methode, 1, (1803), 297–309 (305 fn.).

¹⁷⁶ See also Maarten Bullynck, Vom Zeitalter der formalen Wissenschaften: Anleitung zur Verarbeitung von Erkenntnissen anno 1800, vermittelst einer parallelen Geschichte. (unpublished doctoral thesis, University of Gent, 2006), 261 ff.

¹⁷⁷ See also Maarten Bullynck, Vom Zeitalter der formalen Wissenschaften: Anleitung zur Verarbeitung von Erkenntnissen anno 1800, vermittelst einer parallelen Geschichte. (unpublished doctoral thesis, University of Gent, 2006), 312.

¹⁷⁸ See also Maarten Bullynck, Vom Zeitalter der formalen Wissenschaften: Anleitung zur Verarbeitung von Erkenntnissen anno 1800, vermittelst einer parallelen Geschichte. (unpublished doctoral thesis, University of Gent, 2006), 313.

¹⁷⁹ Ferdinand Rudio, ‘Eine Autobiographie von Gotthold Eisenstein’, Zeitschrift für Mathematik und Physik, 40 (1895), 143–168 (158–159). That Grassmann's theoretical understanding corresponds to Eisenstein's characterisation of ‘essential principle of the new mathematical school’, comes across especially clearly in his ‘General theorem on algebraic curves and surfaces’ formulated in §145 of the A1. Section 4.2 also relates to this.

¹⁸⁰ Felix Klein, Development of mathematics in the 19th century, translated from German by Robert Hermann (Brookline, Massachusetts: Math Sci Press, 1979), 115.

¹⁸¹ See also footnote 51.

¹⁸² This parallel occurs in a particular way in the figure of the outstanding pedagogue Carl Scheibert. He was a devotee of Herbart and was Justus Grassmann's son-in-law. Scheibert considered himself in regard to mathematics in ‘full agreement’ with Justus Grassmann. See Justus Günther Grassmann, Lehrbuch der ebenen und sphärischen Trigonometrie. Für die obern Classen der Gymnasien (Berlin: Reimer, 1835), iv. Scheibert methodically applied Justus Grassmann's paper on number theory to combination theory, developing the latter along Justus Grassmann's lines. Cf. Carl Gottfried Scheibert, Versuch, die Prinzipien der Combinationslehre als einer selbständigen Wissenschaft festzustellen, nebst einer Bezeichnungsmethode in derselben. Programmabhandlung des Stettiner Gymnasiums (Stettin: Effenbart, 1834), 1. And last but not least, Scheibert was also one of those to whom Hermann Grassmann gave private lectures on his extension theory. On the other hand, given the demonstrably crucial influence of his father on him, it is unlikely that Herbart's ideas were passed to Hermann via Scheibert. Also, even if this was the case, it would not be of any great significance. See also Petsche, 2012 (note 88), 215–216.

¹⁸³ Cf. Hermann Grassmann, ‘Lateinischer Lebenslauf Hermann Grassmanns vom 17. Dezember 1831 anlässlich der Anmeldung zur ersten Lehramtsprüfung’, in Petsche et al., 2009 (note 14), 119–135 (126).

¹⁸⁴ Hermann Grassmann, 1833. ‘Lebenslauf Hermann Grassmanns vom 23. März 1833 anlässlich seiner theologischen Prüfung’, in Petsche, 2009 (note 14), 137–149 (145–146).

¹⁸⁵ See Schubring, 1996 (note 23), 61 fn.

¹⁸⁶ See Schubring, 2010 (note 25), 200.

¹⁸⁷ Cf. Victor Schlegel, Hermann Graßmann. Sein Leben und seine Werke (Brockhaus: Leipzig, 1878), 3

¹⁸⁸ Cf. Engel, 1911 (note 4), 16–17.

¹⁸⁹ H. Grassmann, 1831 (note 183), 125; id. 1833 (note 184), 145.

¹⁹⁰ Hermann Grassmann, ‘Was ist wahrhaft wissenschaftlicher Sinn und wodurch erweist sich ein solcher. Abitur-Aufsatz (1827)’, in Petsche et al., 2011 (note 14), 495–497 (495).

¹⁹¹ Cf. Justus Grassmann, Jr., ‘Description of the life of Hermann Grassmann (1877)’, in Petsche et al., 2011 (note 14), 3–8.

¹⁹² J. Grassmann, 1827 (note 141), 1–2 fn.

¹⁹³ J. Grassmann, 1829 (note 92), 184.

¹⁹⁴ Grassmann detailed to an unusual extent his intellectual history in the two previously cited biographical works (see footnotes 183 and 184) from which this information is taken. For further details see Petsche Biography, 2009 (note 15), 15–26.

¹⁹⁵ Engel, 1911 (note 4), 149–154 provides a five-page extract from this letter, in which Grassmann developed his characterisation of the phlegmatic (the persistent), the choleric (the determined), the melancholic (the pleasant), and the sanguine (the open).

¹⁹⁶ H. Grassmann, 1833 (note 184), 146.

¹⁹⁷ Quoted from Engel, 1911 (note 4), 150.

¹⁹⁸ Cf. Friedrich Schleiermacher, Psychologie. Aus Schleiermacher's handschriftlichem Nachlasse und nachgeschriebenen Vorlesungen, edited by Leopold George (Berlin: Reimer, 1862), 307–321.

¹⁹⁹ Arndt, 1986 (note 21), xxix.

²⁰⁰ Arndt, 1986 (note 21), xxxi.

²⁰¹ Cf. A1 (note 1), 20.

²⁰² Lewis, 1977 (note 18), 121.

²⁰³ See Friedrich Schleiermacher, Dialektik. Aus Schleiermacher's handschriftlichen Nachlasse, edited by Ludwig Jonas (Berlin: Reimer, 1839).

²⁰⁴ For the brother, Robert Grassmann, the concept of doubly crossing opposites is also a construction principle of his Gebäudes des Wissens. See Grattan-Guinness, 2011 (note 36). More on this is given in Section 3.4.

²⁰⁵ Cf. Petsche, 2009 (note 15), 244–246.

²⁰⁶ A1 (note 1), 22–23.

²⁰⁷ Cf. A1 (note 1), 22–23.

²⁰⁸ Cf. A1 (note 1), 33–34.

²⁰⁹ Cf. A1 (note 1), 30.

²¹⁰ Cf. A1 (note 1), 30.

²¹¹ Cf. A1 (note 1), 31–32.

²¹² Schubring's objections, to the effect that Schleiermacher's influence on Grassmann was a misunderstanding, and that the A1 was much more profoundly influenced by Grassmann's brother, Robert Grassmann, are refuted in Petsche, 2012 (note 88) in full.

²¹³ See J. Grassmann, 1817 (note 43), x–xi.

²¹⁴ See J. Grassmann, 1817 (note 43), xi.

²¹⁵ J. Grassmann, 1824 (note 43), xv–xvi.

²¹⁶ J. Grassmann, 1824 (note 43), xiiiI–xiv.

²¹⁷ J. Grassmann, 1827 (note 141), 9.

²¹⁸ J. Grassmann, 1829 (note 92), 18–19.

²¹⁹ Friedrich Heinrich Gotthilf Grassmann transferred the whole of the capital from the sale of his school textbooks in 1866 to a foundation for the improvement of the elementary school system in Pomerania. The foundation document contains full and detailed instructions as to how the money was to be used. See Petsche et al., 2009 (note 16), 78–87.

²²⁰ Cf. Jens Brachmann, ‘Über die Möglichkeit, “die Ethik … zerstükkelt mit hervorzubringen.” Die Bestimmung des Verhältnisses von Ethik und “technischen Disciplinen” und die Rezeption erziehungstheoretischer Texte beim frühen Schleiermacher’, in 200 Jahre “Reden über die Religion”. Akten des 1. Internationalen Kongresses der Schleiermacher-Gesellschaft Halle, 14.–17. März 1999, edited by Ulrich Barth and Claus-Dieter Osthövener (Berlin; New York: de Gruyter, 2000), 879–896 (892–893).

²²¹ Cf. Lohmann, 1984 (note 62).

²²² Cf. Petsche, 2009 (note 15), 246–247. In 1810 Bartholdy produced a report ‘On the Order, Methods and Scope of Mathematical and Scientific Teaching’ for the Deputation. This states, inter alia: ‘Without full control of the subjects, he [the teacher – HJP] cannot possibly develop the theorems from inside his listeners and, as must also be the case for presentation of scientific subjects, cause them to invent them from his own resources’. Lohmann, 1984 (note 62), 36–37.

²²³ Cf. Petsche, 2009 (note 15), 246–248.

²²⁴ Schleiermacher 1808 (note 59).

²²⁵ Following the sudden death of Bartholdy in 1815, F.H.G. Grassmann published Bartholdy's linguistic theory (Versuch einer Sprachbildungs-Lehre für Deutsche: Theil 1. Sylbenbildung) in unfinished form, Berlin 1816.

²²⁶ See letter to the publisher Georg Andreas Reimer of 1. 1. 1838, reprinted in Petsche et al., 2009 (note 16), 69–71.

²²⁷ Thus in 1818 J. Grassmann resigned his post of Speaker on account of his reservations against anti-Christian leanings in the Stettin Lodge of the Freemasons. Then in 1823 he became Master of the Lodge. See Adolf Georg Carl Lincke, Geschichte der St. Johannis-Loge Zu den drei Zirkeln (Stettin: R. Grassmann, 1862), 25, 27.

²²⁸ Cf. J. Grassmann's criticism of Kant's view on arithmetic in: J. Grassmann, 1827 (note 141), 12–13. In much the same sense, his son Robert later comments: ‘Kant further asserts that mathematics is grounded in two a priori intuitions […]. This assertion is also founded on an error’. Robert Grassmann, ‘Die Geschichte der Philosophie und erste Einleitung in die Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.1.1. (Stettin: R. Grassmann, 1890), 70.

²²⁹ Quoted from Robert Grassmann, ‘Die Einleitung in die Weltwissenschaften oder Naturwissenschaften oder die physische Propädeutik’, in Robert Grassmann, Das Gebäude des Wissens, Vol. 21.2) (Stettin: R. Grassmann, 1882), 129

²³⁰ J. Grassmann, 1829 (note 92), 174.

²³¹ A comprehensive proof of the erroneousness of the argument that Fries influenced H. Grassmann is given in Petsche, 2012 (note 88).

²³² See Section 3.1

²³³ In the 1840s and 50s, Hermann and his brother Robert went to the Elisenhöhe, which was about an hour away from Stettin. Cf. Engels, 1911 (note 4), 248.

²³⁴ In January 1836 Grassmann came back from Berlin to Stettin where he remained for the rest of his life. One year later he became a teacher at the Stettiner Otto-School. See figure 4.

²³⁵ See Hans-Joachim Petsche, ‘Ernst Abbe's reception of Grassmann in the light of Grassmann's reception of Schleiermacher’, in Hans-Joachim Petsche et al. (eds.), From Past to Future: Grassmann's Work in Context (Basel: Birkhäuser, 2011a), 161–174.

²³⁶ Cf. Grattan-Guinness, 2011 (note 35).

²³⁷ Robert commented that the study of Hegel led him to think ‘that philosophy in its present form […] has no scientific basis and is without scientific method’. He therefore (made) it ‘his life's work to provide philosophy with a basis which would be valid for all time […]. From that time onwards, for the next 45 years, he devoted 4 to 8 hours of hard work every day to this task’. R. Grassmann, 1890 (note 228), xx.

²³⁸ Robert commented that the study of Hegel led him to think ‘that philosophy in its present form […] has no scientific basis and is without scientific method’. He therefore (made) it ‘his life's work to provide philosophy with a basis which would be valid for all time […]. From that time onwards, for the next 45 years, he devoted 4 to 8 hours of hard work every day to this task’. R. Grassmann, 1890 (note 228), xx.

²³⁹ Schleiermacher, 1839 (note 203), 311.

²⁴⁰ A1 (note 1), 23.

²⁴¹ Robert Grassmann, ‘Die Erkenntnislehre. Drittes Buch der Wissenslehre oder Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.2.1. (Stettin: R. Grassmann, 1890), 164.

²⁴² Robert Grassmann, Die Formenlehre oder Mathematik (Stettin: R. Grassmann, 1872), 10.

²⁴³ A2 (note 1), 4.

²⁴⁴ Alfred North Whitehead, A treatise on Universal Algebra. With Applications, vol. 1 (Cambridge: University Press, 1898), x.

²⁴⁵ See Schleiermacher, 1839 (note 203), 291–299.

²⁴⁶ Whilst Hermann Grassmann – presumably influenced by his brother – in his Textbook of Arithmetic (1860) omitted any heuristic considerations, he resumed these a short time later in 1864, in his Textbook of Trigonometry (1865): ‘The whole treatment of the material is different from the one chosen for the Textbook of Arithmetic’, he remarked in the foreword, ‘such that in all of the notes, the leading idea and the whole of further development are emphasised. Thus the heuristic method of the teacher and the deeper delving into the subject by the pupil are considerably facilitated’. Hermann Grassmann, Lehrbuch der Mathematik für höhere Lehranstalten. Zweiter Teil. Trigonometrie (Berlin: Enslin, 1865), vi.

²⁴⁷ See Schleiermacher, 1839 (note 203), 300–312.

²⁴⁸ Robert Grassmann, ‘Die Denklehre. Zweites Buch der Wissenslehre oder Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.1.2. (Stettin: R. Grassmann, 1890), 509.

²⁴⁹ See Robert Grassmann, ‘Die Denklehre. Zweites Buch der Wissenslehre oder Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.1.2. (Stettin: R. Grassmann, 1890), 512.

²⁵⁰ See Robert Grassmann, ‘Die Denklehre. Zweites Buch der Wissenslehre oder Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.1.2. (Stettin: R. Grassmann, 1890), 519.

²⁵¹ See Robert Grassmann, ‘Die Denklehre. Zweites Buch der Wissenslehre oder Philosophie’, in Robert Grassmann, Das Gebäude des Wissens. Vol. 1.1.2. (Stettin: R. Grassmann, 1890), 520.

²⁵² Grattan-Guinness, 2009 (note 35), 33.

²⁵³ A1, 26. The scheme has been investigated multiple times in the literature. See for example Lewis, 1977 (note 18), 124--127; Radu, 2000 (note 29), 18--21; Petsche Biography, 2009 (note 15), 235--241.

²⁵⁴ See for example Volker Peckhaus, ‘The influence of Hermann Günther Grassmann and Robert Grassmann on Ernst Schröder's algebra of logic’, in Schubring, 1996 (note 22), 217–227; id., ‘Robert and Hermann Grassmann's Influence on the History of Formal Logic’, in Petsche et al., 2011 (note 14), 221–228.

²⁵⁵ ‘This latter form of development is able to move more freely’, Robert Grassmann stressed, ‘it is able to examine and throw light on the various possible routes, as well as the ideas on which development is founded; hence the reader is in turn more encouraged for his own part to perform experiments in the development of formulae and proofs in various directions…[In] representation in the free development of thought, development [is] led by observations […] as to how one can arrive at a rigorous formula and how one can modify the latter…The advantage of this representation is that one constantly weighs up which routes to take, and that one can throw critical light on the route one has embarked on. Each of these two types of representation complements the other, and each will attract its own devotees’. Robert Grassmann, ‘Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formel-Entwicklung’, in Robert Grassmann, Die Formenlehre oder Mathematik in strenger Formelentwicklung (Stettin: R. Grassmann, 1891), viii.

²⁵⁶ ‘This latter form of development is able to move more freely’, Robert Grassmann stressed, ‘it is able to examine and throw light on the various possible routes, as well as the ideas on which development is founded; hence the reader is in turn more encouraged for his own part to perform experiments in the development of formulae and proofs in various directions…[In] representation in the free development of thought, development [is] led by observations […] as to how one can arrive at a rigorous formula and how one can modify the latter…The advantage of this representation is that one constantly weighs up which routes to take, and that one can throw critical light on the route one has embarked on. Each of these two types of representation complements the other, and each will attract its own devotees’. Robert Grassmann, ‘Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formel-Entwicklung’, in Robert Grassmann, Die Formenlehre oder Mathematik in strenger Formelentwicklung (Stettin: R. Grassmann, 1891), vi.

²⁵⁷ ‘This latter form of development is able to move more freely’, Robert Grassmann stressed, ‘it is able to examine and throw light on the various possible routes, as well as the ideas on which development is founded; hence the reader is in turn more encouraged for his own part to perform experiments in the development of formulae and proofs in various directions…[In] representation in the free development of thought, development [is] led by observations […] as to how one can arrive at a rigorous formula and how one can modify the latter…The advantage of this representation is that one constantly weighs up which routes to take, and that one can throw critical light on the route one has embarked on. Each of these two types of representation complements the other, and each will attract its own devotees’. Robert Grassmann, ‘Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formel-Entwicklung’, in Robert Grassmann, Die Formenlehre oder Mathematik in strenger Formelentwicklung (Stettin: R. Grassmann, 1891), viii.

²⁵⁸ A1 (note 1), 20.

²⁵⁹ Sophus Lie, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, edited by Georg Scheffers (Leipzig: Teubner, 1893), 618.

²⁶⁰ A1 (note 1), 27.

²⁶¹ J. Grassmann, 1827 (note 141), 19.

²⁶² J. Grassmann, 1827 (note 141), 21.

²⁶³ J. Grassmann, 1827 (note 141), 18.

²⁶⁴ J. Grassmann, 1827 (note 141), 4.

²⁶⁵ J. Grassmann, 1827 (note 141), 4.

²⁶⁶ J. Grassmann, 1829 (note 92), 34.

²⁶⁷ Cf. J. Grassmann, 1829 (note 92), 34., 34–35.

²⁶⁸ Cf. J. Grassmann, 1829 (note 92), 34., 35.

²⁶⁹ A1 (note 1), 158.

²⁷⁰ A1 (note 1), 48.

²⁷¹ John Brown, ‘Grassmann, geometry and mechanics’, in Petsche et al., 2011 (note 14), 287–302 (289).

²⁷² In the same manner, Grassmann initially derived this in his examination essay on tides. See Hermann Grassmann, ‘Theorie der Ebbe und Flut. Prüfungsarbeit von 1840’, edited by Justus Grassmann d. J., in Hermann Grassmann, Gesammelte mathematische und physikalische Werke. Vol. 3.1. (Leipzig: Teubner, 1911), 1–238 (30).

²⁷³ Whilst he continued to call it a mixed product in the A1, the reference in the A2 was simply to ‘the product with respect to a principal domain’. A2 (note 1), 65.

²⁷⁴ Cf. Bertfried Fauser, ‘Grade Free Product Formulae from Grassmann-Hopf Gebras’ in Rafal Ablamowicz (ed.), Clifford Algebras. Applications to Mathematics, Physics, and Engineering (Boston: Birkhäuser, 2004), 279–303 (290).

²⁷⁵ A2 (note 1), 7.

²⁷⁶ The inner product, which was an addition to the extension theory of 1862, represents a further necessary complementary product formation, albeit of a different nature.

²⁷⁷ A1 (note 1), 245.

²⁷⁸ A1 (note 1), 245.

²⁷⁹ Cf. Felix Klein, Development of mathematics in the 19th century, translated from the German by Robert Hermann (Brookline, Massachusetts: Math Sci Press, 1979), 168.

²⁸⁰ Cf. Felix Klein, Development of mathematics in the 19th century, translated from the German by Robert Hermann (Brookline, Massachusetts: Math Sci Press, 1979), 169.

²⁸¹ Quoted from a letter, which Engel received from Klein. In Engel, 1911 (note 4), 312.

²⁸² Reprinted in Hermann Grassmann. Gesammelte mathematische und physikalische Werke. Vol. 2.1. Eds. Eduard Study, Georg Scheffers and Friedrich Engel (Leipzig: Teubner, 1904), 49–238.

²⁸³ A1 (note 1), 245.

²⁸⁴ Hans Wußing, ‘Externalismus – Internalismus’, N.T.M., 15 (2007), 284–288 (287).