Hermann Graßmann pp 165-220 | Cite as

# Hermann Günther Graßmann’s contributions to the development of mathematics and their place in the history of mathematics

Chapter

## Abstract

In order to do justice to the demands voiced by the exceptional German mathematician Alfred Clebsch, one will have to grasp the significance and the fate of Hermann Günther Graßmann’s mathematical oeuvre against the backdrop of the revolutionary changes that occurred in geometric ideas in the first half of the 19^{th} century.

## Keywords

Projective Geometry Algebraic Curf Basic Series Extension Theory Exterior Algebra
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## Notes

- 1.Clebsch 1871, p. 3.Google Scholar
- 2.See Wußing 1984, p. 25.Google Scholar
- 3.First steps towards an analytical geometry can be traced via E. Torricelli, G. Galilei, J. Kepler, N. Oresme to Appolonius of Perga. — See Böhm et. al. 1975, p. 16sqq.Google Scholar
- 4.See Struik 1987, 96sqq.Google Scholar
- 5.See Clebsch 1871, p. 10.Google Scholar
- 6.See Clebsch 1871, p. 10.Google Scholar
- 7.Leibniz 1686, p. 22.Google Scholar
- 8.In 1890 F. Engel noted that in analytical geometry something “unsteady and irregular” came to light, “for the single steps of the calculation almost never make any geometrical sense, they usually seem like a sleight of hand...” (Engel 1890, p. 17/18).Google Scholar
- 9.See Wußing 1984, p. 35sqq.Google Scholar
- 11.See Study 1898, p. 159.Google Scholar
- 12.See Loria 1888, p. 115sqq.Google Scholar
- 13.See Alexandroff /Markuschewitsch /Chintschin 1971, p. 342sqq, and
*History of science*, vol. 3 1965, p. 34sq.Google Scholar - 14.It was Cayley who coined the term “
*n*-dimensional geometry”. — See Alexandroff / Markuschewitsch / Chintschin 1971, p. 342sqq.Google Scholar - 15.Lie 1934, p. 105.Google Scholar
- 16.See Licis 1976, p. 87sqq.Google Scholar
- 17.See Wußing 1976, p. 53sqq.Google Scholar
- 18.See Ruzavin 1977, p. 99sq., also Helmholtz 1868, Riemann 1876a, and Klein 1921.Google Scholar
- 19.See Blaschke 1948, p. 12sqq., and Wußing 1984, p. 27sq. These authors are also relevant in the following lines.Google Scholar
- 20.See Wußing /Arnold 1975, p. 270sqq.Google Scholar
- 22.See Fano 1907.Google Scholar
- 23.Wußing 1984, p. 26.Google Scholar
- 24.See ibidem.Google Scholar
- 25.Ibidem, p. 33.Google Scholar
- 28.Quoted from Wußing’s introduction to Klein 1974, p. 20.Google Scholar
- 29.See Gerhardt 1877, p. 289sq., and Struik 1987, p. 165.Google Scholar
- 33.EBBE, p. 11.Google Scholar
- 34.Quoted from Struik 1987, p. 135.Google Scholar
- 35.EBBE, p. 17.Google Scholar
- 36.More information on “Graßmann’s or exterior algebra” and similar concepts from the viewpoint of recent mathematics can be found in: Naas /Schmid 1972a, p. 130, 648; Naas/ Schmid 1972b, p. 209sq.; Kupcov 1977; Oniščik 1977a; Oniščik 1977b; Eisenreich 1971, p. 40sqq.; Gröbner 1966, p. 30sqq.; Groh 1956; Schatz 1970.Google Scholar
- 37.See EBBE, p. 19/20.Google Scholar
- 38.See ibidem, p. 28.Google Scholar
- 39.See ibidem, p. 30.Google Scholar
- 40.See ibidem, p. 40.Google Scholar
- 41.See ibidem p. 56sqq.Google Scholar
- 42.EBBE, p. 57.Google Scholar
- 43.EBBE, p. 81.Google Scholar
- 44.See Justus Graßmann’s, Hermann Graßmann’s son’s, remark concerning the dissertation. In GW31, p. 221.Google Scholar
- 47.In his equipollent calculus (Bellavitis 1835) Bellavitis developed a way of calculating with equipollent displacements, defining two displacements as equipollent when they are identical in length, direction and orientation, but not in situation. In closer examinations he found vector addition, among other insights. — See Rothe 1916.Google Scholar
- 48.See, for more details, Becker /Hofmann 1951, p. 326sqq.Google Scholar
- 49.Klein 1979, p. 161.Google Scholar
- 50.Biermann 1973, p. 38.Google Scholar
- 51.Wußing /Arnold 1975, p. 352.Google Scholar
- 52.See also Clebsch 1871, p. 15sq., and Clebsch 1874, p. 12.Google Scholar
- 54.Klein 1979, p. 162.Google Scholar
- 55.Clebsch 1871, p. 8.Google Scholar
- 57.See Graßmann’s general elaborations in A1, p. 33–43.Google Scholar
- 58.A1, p. 40.Google Scholar
- 59.A1, p. 42.Google Scholar
- 61.See A1, p. 45 and 154.Google Scholar
- 63.See A1, p. 48.Google Scholar
- 64.A1, p. 47.Google Scholar
- 65.F. Engel, the editor of Graßmann’s collected works, makes this comment on the 1844
*Extension Theory in GW11*, p. 404.Google Scholar - 66.A1, p. 50.Google Scholar
- 67.Riemann 1876a, p. 257. English translation taken from Smith 1929.Google Scholar
- 68.Riemann 1876a, p. 255.Google Scholar
- 69.A1, p. 58.Google Scholar
- 70.See Segre 1912, p. 22/23.Google Scholar
- 71.A1, p. 59.Google Scholar
- 72.A1, p. 60.Google Scholar
- 73.A1, p. 62.Google Scholar
- 74.Ibidem.Google Scholar
- 75.A1, p. 62/63.Google Scholar
- 76.See, among other sources, Helmholtz 1868.Google Scholar
- 77.Helmholtz 1868, p. 49.Google Scholar
- 78.Ibidem p. 32.Google Scholar
- 79.A1, p. 73. — See also my presentation of Graßmann’s ideas concerning the essence of the mathematical method in chapter 4, section 4.Google Scholar
- 81.In this context, Klein speaks directly of the “Grassmann Determinant principle for the plane” and the “Grassmann principle for space”. See Klein 1939, p. 21sqq., 29sqq.Google Scholar
- 82.See A1, p. 91–93.Google Scholar
- 84.See A1, 113sqq.Google Scholar
- 85.A1, 133/134.Google Scholar
- 86.Enriques 1907, p. 53.Google Scholar
- 87.Graßmann writes: “Instead of shadowing the result of a fundamental conjunction [vector addition, exterior multiplication, as well as the corresponding inverse operations — H.-J. P.], one can shadow its terms in the same sense.” (A1, p. 144).Google Scholar
- 88.A1, p. 157.Google Scholar
- 89.A1, p. 173.Google Scholar
- 90.A2, p. 144.Google Scholar
- 91.See A1, p. 172–179.Google Scholar
- 92.A1, p. 186.Google Scholar
- 93.See Clebsch 1871, p. 28.Google Scholar
- 94.See Wußing 1984, p. 43.Google Scholar
- 95.See Clebsch 1871, p. 28 (footnote).Google Scholar
- 96.See A1, p. 197sq.Google Scholar
- 97.See also the explanations in Rothe 1916, p. 1284sqq.Google Scholar
- 98.Later, Graßmann became aware how unfruitful the level of extreme abstraction was when it came to introducing the regressive product. For this reason, when in 1877 he presented the second edition of the 1844
*Extension Theory*, he added an appendix to this section, in which he gave a simpler and more accessible definition of the regressive product. See A1, p. 295.Google Scholar - 99.See A1, p. 202Google Scholar
- 100.A1, p. 207.Google Scholar
- 101.A1, p. 235.Google Scholar
- 102.Ibidem.Google Scholar
- 103.A1, p. 236.Google Scholar
- 104.See Graßmann’s investigations in H. Graßmann 1851b.Google Scholar
- 105.See in this context Scheffer’s remark in GW21, p. 393sqq.Google Scholar
- 106.Klein 1928, p. 132.Google Scholar
- 107.Ibidem.Google Scholar
- 108.See Bloch 1951.Google Scholar
- 110.A1, p. 233.Google Scholar
- 111.These are the works by H. Graßmann 1846, 1848a, 1851a-c, 1852, 1855a-e, 1855g.Google Scholar
- 112.Cremona 1860, p. 356sq.Google Scholar
- 113.Möbius 1827, p. X.Google Scholar
- 114.A1, p. 248.Google Scholar
- 115.See Klein 1927, p. 10sqq.Google Scholar
- 116.See A1, p. 252.Google Scholar
- 117.Klein 1927, p. 11/12. This section of the text is not included in the English translation of Klein’s
*Development of Mathematics in the 19th Century*(Klein 1979).Google Scholar - 118.A1, p. 255.Google Scholar
- 119.See A1, p. 252sq.Google Scholar
- 120.Concerning the Graßmann cross ratio, See Blaschke 1948, p. 95sq., and Keller 1963, p. 235sqq.Google Scholar
- 121.Möbius, who had worked intensively on similar problems, became aware of this connection, which he had used indirectly, through this work of Graßmann. — See Engel’s remark concerning the 1844
*Extension Theory*in GW11, p. 411sqq.Google Scholar - 122.See A1, p. 271.Google Scholar
- 124.See A1, p. 273sqq.Google Scholar
- 125.Lotze 1929, p. 79sqq., is one example where the connection between Graßmann’s open products and tensors is shown.Google Scholar
- 126.The original French passage from Leibniz’ letter is reprinted in GW11, p. 417–420.Google Scholar
- 127.Leibniz in a letter to Christian Huygens (1629–1695) dated 8 September 1679. In: Leibniz 1976, p. 248–258 (p. 250). See also GW11, p. 418. A German translation of this passage, lacking bibliographical references, can be found in Bell 1967, p. 129.Google Scholar
- 128.PREIS, p. 318.Google Scholar
- 129.See Graßmann’s letter to Hankel, 2 February 1867. The corresponding passage is reproduced in BIO, p. 110, footnote.Google Scholar
- 130.See also Engel’s remark in GW11, p. 421.Google Scholar
- 131.PREIS, p. 320. So this amounts to saying that the following theorem is not universally valid: (
*a c***8***b c*) Λ (*a c d***8***a c d*) a (*a c d***8***b c d*).Google Scholar - 132.See PREIS, p. 321.Google Scholar
- 133.PREIS, p. 332.Google Scholar
- 134.PREIS, p. 334.Google Scholar
- 135.PREIS, p. 333.Google Scholar
- 136.PREIS, p. 336.Google Scholar
- 137.PREIS, p. 339.Google Scholar
- 138.PREIS, p. 340.Google Scholar
- 139.See also Engel’s remark in GW11, p. 421.Google Scholar
- 140.Here, Couturat’s appreciative reaction to Graßmann’s
*Geometric Analysis*must be mentioned. — See Couturat 1969, p. 529–538.Google Scholar - 142.A2, xiii/xivGoogle Scholar
- 143.See Wußing’s remarks in Wußing 1984, p. 239.Google Scholar
- 144.With these inquiries, Graßmann partially went beyond the results which Weierstraß and Jordan reached six and eight years later. See Bourbaki 1971, p. 109.Google Scholar
- 146.Klein 1979, p. 165. In the original, the last word of this quote reads’ space’, a translation error.Google Scholar
- 147.ZL, p. 3.Google Scholar
- 148.ZL, p. 4.Google Scholar
- 149.ZL, p. 6.Google Scholar
- 150.A1, p. 23.Google Scholar
- 151.A1, p. 24.Google Scholar
- 152.A1, p. 25.Google Scholar
- 153.A1, p. 23 (footnote).Google Scholar
- 154.LA, p. V.Google Scholar
- 155.Ibidem.Google Scholar
- 156.This is how A. Heintze remembers his days as a pupil of Justus Graßmann: “Also in mathematics we were given no problems in written form, and that wasn’t good. We lost the habit of solving them, and when a paper had to be written, we mostly wrote it in an unorganized fashion, even copying from one another, which caused great trouble for Graßmann.” (Heintze 1907, p. 44).Google Scholar
- 157.Ibidem.Google Scholar
- 159.LA, p. 3.Google Scholar
- 160.LA, p. 17.Google Scholar
- 161.LA, p. 21.Google Scholar
- 162.Interestingly, Helmholtz later turned to Graßmann’s arithmetic in order to discuss under which conditions magnitudes in physics may be conceptualized as named numbers. He put the problem this way: “In which objective sense may we express the relations of real objects by named numbers as magnitudes, and under which conditions may we do so?” (Helmholtz 1887, p. 304).Google Scholar
- 163.See § 7 Division in: LA, p. 45sqq.Google Scholar
- 165.The first of these two sentences is missing from Lloyd Kannenberg’s translation of A2 (p. 3.). See also GW12, p. 12.Google Scholar
- 166.A2, p. 4.Google Scholar
- 167.R. Graßmann 1890c, S. VI.Google Scholar
- 168.This is also the case in Lewis 1995 and the noteworthy reflections by Radu 2000, p. 205.Google Scholar
- 169.Schlegel 1878, p. 42.Google Scholar
- 170.The following passages are analyzed in detail in Radu 2000, p. 205sqq.Google Scholar
- 171.See LA, p. 3.Google Scholar
- 172.See Hao Wang 1957, p. 147.Google Scholar
- 173.LA, p. 4.Google Scholar
- 174.Radu discusses at length that Graßmann’s explanation (LA, p. 4): “
*a*+(*b+e*)=*a+b+e*” is not constructively justified (2000, p. 216sqq).Google Scholar - 175.When Schleiermacher remarked: “So the identity of the process and the immutability of the relation between thought and object are the two fundamental aspects of knowledge.” (Schleiermacher 1942, p. 130), this probably was a strong reason for Graßmann to accept the validity of recursive proofs and the “pre-logical” characteristics of the complete induction’s validity.Google Scholar
- 176.LA, p. 16. (Italics added.)Google Scholar
- 177.LA, p. 17.Google Scholar
- 178.LA, p. 18.Google Scholar
- 179.See LA, p. 19.Google Scholar
- 180.Hao Wang 1957, p. 148.Google Scholar
- 181.Hao Wang 1957, p. 147.Google Scholar
- 182.See also Radu 2000, p. 214.Google Scholar
- 183.LA, p. 3.Google Scholar
- 184.See R. Graßmann 1890f., p. 3–7.Google Scholar
- 185.See Zahn 1874, p. 583sqq.Google Scholar
- 186.See Bourbaki 1971, p. 33/34.Google Scholar
- 187.Hankel 1867, p. 1–34.Google Scholar
- 188.Wußing 1984, p. 240 (footnote 224).Google Scholar
- 189.See Bourbaki 1971, p. 33, 34, 71.Google Scholar
- 190.For more details, see Wußing 1984 and Bourbaki 1971.Google Scholar
- 192.Hankel 1867, p. 12.Google Scholar
- 193.Ibidem, p. 13.Google Scholar
- 194.Ibidem, p. 16Google Scholar
- 195.Hao Wang emphasizes: “It is rather well-known, through Peano’s own acknowledgement..., that Peano borrowed his axioms from Dedekind and made extensive use of Graßmann’s work in his development of the axioms.” (Hao Wang 1957, p. 145). — See also Wußing 1984, p. 239.Google Scholar
- 196.Letter from F. Klein to F. Engel, 21 January 1911. In: BIO, p. 312.Google Scholar
- 197.See Klein 1974, p. 55, 58sq.Google Scholar
- 198.See Klein 1974, p. 68.Google Scholar
- 199.See ibidem, p. 71.Google Scholar
- 200.Klein 1921, p. 320.Google Scholar
- 201.Wußing 1969, p. 143. See also Klein’s thorough appreciation of Graßmann’s work and his concomitant explication of affine and projective transformations in
*Elementary Mathematics*(“Elementarmathematik”, Klein 1925).Google Scholar - 202.See Ruzavin 1977, p. 59, and Wußing 1984, p. 240 (footnote).Google Scholar
- 204.See Severi 1916. See also Burau 1953.Google Scholar
- 205.Wußing 1984, p. 238.Google Scholar
- 206.See ibidem, p. 239sqq.Google Scholar
- 207.Dyck 1882, p. 2.Google Scholar
- 208.Ibidem. See also in Dyck 1882. p. 43 (footnote).Google Scholar
- 209.Whitehead 1898, p. 32.Google Scholar
- 210.Whitehead 1898, p. v.Google Scholar
- 211.Ibidem, p. x.Google Scholar
- 213.In the context of the theory of algorithms, Birjukova and Birjukov (1997) point out that Graßmann’s “general theory of forms” (A1 1844) already gave a definition for the concept of the abstract group (10 years before Cayley) and of the ring (70 years before Fraenkel). This fact has so far gone unnoticed.Google Scholar
- 214.See in this context M. J. Crowe’s detailed analysis in
*A History of Vector Analysis*(1994).Google Scholar - 216.See the terminology chosen by A. Lotze in Lotze 1929. Burali-Forti 1921, p. 239–244, gives an instructive overview of the different concepts used up until 1920.Google Scholar
- 217.Polak, in his Hamilton-biography 1993, p. 233–236, tries to show that Peano, in his
*Calcolo geometrico secondo l’*Ausdehnungslehre*di H. Graßmann*(1888), does not actually follow Graßmann, but Gibbs, who had composed two small treatises on the*Elements of vector analysis*in 1881 and 1884, passing on to friends, among them G. Basso, Peano’s teacher, and J. Lüroth, a friend of Peano, 130 exemplars of these texts. This amounts to saying that it was Gibbs, who according to Polak had developed vector algebra mostly on his own, is at the beginning of the modern theory of affine vector spaces and that Graßmann had no real effect in this sense (see the overview in Zaddach 1994, p. 11). But Polak’s argument, which relies especially on the clarity of Peano’s thinking and Graßmann’s obscurity in the corresponding passages, is not a convincing one. See in this context the Peano-biography by Kennedy 2002.Google Scholar - 218.Bourbaki 1998, p. 66.Google Scholar
- 219.See Klein 1939, p. 51sqq.Google Scholar
- 221.See also Bourbaki 1961, p. 140.Google Scholar
- 222.See, as an example, Gröbner 1966, p. 7.Google Scholar
- 223.See Pillis 1968.Google Scholar
- 224.See for example Eisenreich 1971, p. 132sqq.Google Scholar
- 225.See in this context Groh 1956.Google Scholar
- 226.See Cartan 1952, p. 38/39, and Cartan 1953, p. 241/242.Google Scholar
- 227.See Bourbaki 1971, p. 83sq.Google Scholar

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