- Klein , F. 1967 . Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert , Vol. 1 , 175 – 175 . New York : Dover . reprint of original 1926 edition),
- Boyer , C. 1968 . A history of mathematics , 584 – 584 . New York : Wiley .
- Kline , M. 1972 . Mathematical thought from ancient to modern times , 782 – 782 . New York : Oxford University Press .
- Grassmanns Leben constitutes vol. 3, pt. 2 of Hermann Grassmanns gesammelte mathematische und physikalische Werke B. G. Teubner Leipzig 1894–1911 3 vols. in 6 parts, general editor F. Engel, with assistance from J. Lüroth, E. Study, Justus Grassmann (eldest son of Hermann), Hermann Grassmann Junior, and G. Scheffers.
- 1967 . A history of vector analysis: the evolution of the idea of a vectorial system , Notre Dame : University of Notre Dame .
- Henrici , O. 1910–1911 . “ Geometry II. Projective geometry ” . In Encyclopaedia Britannica , 11th ed. Vol. 11 , 688 – 706 . (p. 688).
- That projective geometry was a major stimulus towards a revision of the philosophy of pure mathematics is the thesis of E. Nagel in ‘The formation of modern conceptions of formal logic in the development of geometry’, Osiris 1939 7 142 224
- Steiner , J. 1832 . Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander Berlin Th. 1, vi; as translated by W. C. Swabey and M. C. Swabey in E. Cassirer, Substance and function (New York: Dover, 1953, reprint of 1923 edition), 78.
- ‘Zur Metaphysik der Mathematik’, Gauss C.F. Werke 1929 12 57 57
- Letter to Olbers (1817), Werke 1900 8 177 177
- Körner , S. 1960 . The philosophy of mathematics: an introduction , 183 – 183 . New York : Harper .
- 1866 . Traité des propriétés projectives Vol. 2 , 357 – 357 . Paris (The first edition of 1822 contains Cauchy's ‘Rapport’.)
- 1866 . Traité des propriétés projectives Vol. 2 , 5 – 5 . Paris
- A1, 15. Reference to the A1 is to the 1878 reprint as it appears in Werke 1 pt. 1. Spaced (gesperrte) print in the A1 appears in translated quotations in this paper as italic print. (See footnote 132 on the use of italics in the A1). Square brackets enclose: my insertions as translator and interpreter; a quotation from the German text; or new figure numbers replacing those in the A1. (See footnote 155 on the use of square brackets in Grassmann's mathematical notation.)
- Ohm , M. 1819 . Kritische Beleuchtungen der Mathematik überhaupt und der Euklidischen Geometrie insbesondere 7 – 7 . Berlin
- Bolzano , B. 1843 . Versuch einer objektiven Begründung der Lehre von den drei Dimensionen des Raumes . Abh. Königl. Böhm. Gesell. Wiss. , 3 ( 5 ) : 201 – 215 . publ. 1845 (p. 203)
- 1939 . Osiris , 7 : 173 – 173 .
- Werke, vol. 3, pt. 2, 101, as translated in Osiris 1939 7 173 173
- 1883 . Kant's prolegomena and metaphysical foundations of natural science , London : G. Bell . translated by E. B. Bax;
- Encyklopädie der Elementar Mathematik Vol. 2 , 146 – 146 . bk. 1, sect. 2 as quoted in Cassirer (footnote 8), 106
- Encyklopädie der Elementar Mathematik Vol. 2 , 145 – 145 .
- 1969 . The world as will and representation , Vol. 1 , 73 – 73 . New York : Dover . (translated by E. F. J. Payne
- Körner . 1960 . The philosophy of mathematics: an introduction , 28f – 28f . New York : Harper . G. Buchdahl, Metaphysics and the philosophy of science (Cambridge, Mass.: The MIT Press, 1969), 580, 611ff.
- The ‘Lebenslauf’ is one of the records described by F. Engel in the Foreword to his account of Grassmann's life and work Werke vol. 3 viii viii pt. 2 as having been found in Stettin, Grassmann's home town. It may have been written sometime in the period of the two theological examinations, taken in 1834 and 1839, with which it was found.
- Werke , 3 21 – 22 . pt. 2
- Dialektik Friedrich Schleiermacher's sämmtliche Werke G. Reimer Berlin 1839 (Reimer, a friend of Schleiermacher's, also published J. Grassmann's Raumlehre, 1824 (footnote 98 below) and Trigonometrie, 1835.) Only in the twentieth century have other editions of the Dialektik been produced, but in spite of its recognized organizational difficulties and obscurities this Jonas edition is still generally regarded as the standard. It is cited by page number in later footnotes.
- Werke , 3 91 – 91 . pt. 2 131, 133
- Schlegel , V. 1878 . Hermann Grassmann. Sein Leben und seine Werke , 14 – 14 . Leipzig : Brockhaus .
- Letter to St.-Venant, 18 Werke 1847 April 3 42 42 pt. 2
- Brandt , R.B. 1941 . The philosophy of Schleiermacher: the development of his theory of scientific and religious knowledge , New York : Greenwood . reprinted 1968), 204–205; H. L. Friess, Schleiermacher's soliloquies (Chicago: Open Court Press, 1926), xxvii, 12.
- ‘Actual, specific’ Spiegler G. The eternal covenant: Schleiermacher's experiment in cultural theology Harper & Row New York 1967 82 82
- Dialektik , 309 – 309 .
- Two works in English have been relied upon as guides to Schleiermacher's work Spiegler G. The eternal covenant: Schleiermacher's experiment in cultural theology Harper & Row New York 1967 82 82 and R. B. Brandt (footnote 31). Also of use was W. Dilthey, Leben Schleiermachers, Zweiter Bd., compiled by M. Redeker, Bd. 14 of Wilhelm Dilthey, Gesammelte Schriften (Göttingen: 1966).
- In Sämmtliche Werke G. Reimer Berlin 1846 1
- In Sämmtliche Werke G. Reimer Berlin 1846 1 333 338 div. 3
- In Sämmtliche Werke G. Reimer Berlin 1846 1 336 336 div. 3
- ‘Positing particular skeptical hypotheses’ Spiegler The eternal covenant: Schleiermacher's experiment in cultural theology Harper & Row New York 1967 46 46
- Grundlinien , 336 – 337 .
- Grundlinien , 339 – 340 .
- Spiegler , G. 1967 . The eternal covenant: Schleiermacher's experiment in cultural theology , 53 – 53 . New York : Harper & Row .
- Dialektik , 312 – 312 .
- Dialektik , 39 – 39 .
- Dialektik , 43 – 43 .
- Dialektik , 48 – 48 .
- Dialektik , 48 – 48 .
- Dialektik , 140 – 140 . 335–336
- Dialektik , 336 – 336 .
- Dialektik , 336 – 336 .
- Dialektik , 50 – 50 .
- Dialektik , 51 – 51 .
- Dialektik , 68 – 69 .
- Dialektik , 69 – 69 .
- Dialektik , 490 – 490 .
- Dialektik , 69 – 69 .
- Dialektik , 179 – 179 .
- Dialektik , 178 – 178 .
- Dialektik , 179 – 180 .
- Dialektik , 285 – 288 .
- Dialektik , 287 – 288 .
- Dialektik , 291 – 292 .
- Dialektik , 299 – 299 .
- Dialektik , 299 – 299 .
- Dialektik , 296 – 296 .
- Dialektik , 296 – 296 .
- Dialektik , 296 – 296 .
- Dialektik , 297 – 297 .
- Dialektik , 557 – 557 . See Brandt (footnote 31), 208–209 for further discussion on species-genus determination in the Dialektik.
- 1929 . Immanuel Kant's Critique of pure reason , New York : St. Martin's Press . translated by N. K. Smith 1965
- 1929 . Immanuel Kant's Critique of pure reason , 547 – 548 . New York : St. Martin's Press . translated by N. K. Smith 1965
- Dialektik , 299 – 299 .
- Dialektik , 303 – 303 .
- Dialektik , 300 – 303 . Dilthey (footnote 34), 224
- Dialektik , 309 – 309 .
- Dialektik , 92 – 92 .
- Dialektik , 309 – 309 .
- Dialektik ,
- Dialektik , 288 – 288 .
- Dialektik , 59 – 59 .
- Dialektik , 310 – 310 .
- Dialektik , 311 – 311 .
- Dialektik ,
- Dialektik , 61 – 63 .
- 1829 . Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft , 172 – 172 . Stettin : Morin .
- Werke , 1 404 – 404 . pt. 1
- A1 22 – 23 . The following remarks on my translations from the A1 are necessary. Grassmann achieved more recognition during his lifetime as a philologist than as a mathematician, and at least a trace of Grassmann as philologist appears in the A1 in his preference for simple German over Latin or other foreign terms. ‘Strecke’ and ‘Stufe’ are two such terms used for two of the key concepts later in the A1. I think it is important to signalize the fact that their meanings were unique to Grassmann—at least for him, it seems clear, they were new mathematical concepts. For this reason, ‘stretch’ is chosen to translate ‘Strecke’, since a modern reader would probably have no strong mathematical preconceptions about it and its usual meaning of a continuous extent of length suits the geometrical prototype of ‘Strecke’. (This translation was suggested to me by Prof. Dr. C. Thiel.) For analogous reasons ‘step’ is used to translate ‘Stufe’. My translation tends to follow the German closely, and I believe that by using such words as ‘stretch’ and ‘step’, which have no wide currency in mathematics, Grassmann's effect of novelty in a mathematical context is achieved.
- A1 23 – 23 .
- A1 23 – 23 .
- Grassmann , R. 1872 . Die Formenlehre oder Mathematik Stettin reprinted Hildesheim: Georg Olms, 1966), pt. II: Die Begriffslehre oder Logik.
- A1 24 – 25 .
- A1 25 – 25 .
- A1 25 – 25 .
- A1 25 – 25 .
- A1 25 – 26 .
- 1817 . Raumlehre für Volkschulen. Erster Theil: Ebene räumliche Verbindungslehre , x – xi . Berlin : In der Realschulbuchhandlung .
- 1824 . Raumlehre für die untern Klassen der Gymnasien, und für Volkschulen. Zweiter Theil. Ebene räumliche Grössenlehre , 194 – 195 . Berlin : G. Reimer .
- A1 26 – 27 .
- A1 28 – 28 .
- Grassmann , J. 1829 . Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft , 33 – 34 . Stettin : Morin .
- A1 28 – 29 .
- Cassirer . 1832 . Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander 78 – 78 . Berlin
- Cassirer . 1832 . Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander 99 – 99 . Berlin
- Other works by Cassirer in which the A1 is cited are: The philosophy of symbolic forms Yale University Press New Haven 1953 3 The phenomenology of knowledge, 352; and The problem of knowledge: philosophy, science, and history since Hegel (New Haven: Yale University Press, 1950), 49.
- Cassirer . 1832 . Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander 111 – 111 . Berlin
- Such a misconception could come about from the association of the A1 with Peano and Whitehead, who in turn are associated with the axiomatic development. But there is no evidence that the A1 influenced these two in this way, and neither Peano nor Whitehead discussed the larger import of the A1 in the development of mathematics. (On Grassmann and Whitehead see V. Lowe A.N. Whitehead on his mathematical goals: a letter of 1912 Annals of science 1975 32 85 101
- A1 30 – 32 .
- A1 46 – 46 .
- Kant . 1929 . Immanuel Kant's Critique of pure reason , 577 – 577 . New York : St. Martin's Press . translated by N. K. Smith
- Kant . 1929 . Immanuel Kant's Critique of pure reason , 577 – 577 . New York : St. Martin's Press . translated by N. K. Smith
- Kant . 1929 . Immanuel Kant's Critique of pure reason , 577 – 577 . New York : St. Martin's Press . translated by N. K. Smith
- Kant . 1929 . Immanuel Kant's Critique of pure reason , 578 – 578 . New York : St. Martin's Press . translated by N. K. Smith
- Schopenhauer . 1969 . The world as will and representation , Vol. 1 , 70 – 70 . New York : Dover . (translated by E. F. J. Payne
- Schopenhauer . 1969 . The world as will and representation , Vol. 1 , 52 – 52 . New York : Dover . (translated by E. F. J. Payne
- Schopenhauer . 1969 . The world as will and representation , Vol. 1 , 71 – 71 . New York : Dover . (translated by E. F. J. Payne
- Schopenhauer . 1969 . The world as will and representation , Vol. 1 , 53 – 58 . New York : Dover . (translated by E. F. J. Payne
- A1 23 – 24 .
- 1889 . The Open Court , 2 : 1463 – 1464 . Carus (1852–1919) was a student of H. Grassmann in Stettin
- A1 63 – 64 .
- A1 65 – 65 .
- A1 66 – 66 .
- A1 33 – 33 .
- A1 28 – 28 .
- A1 33 – 34 .
- A1 34 – 35 .
- A1 35 – 35 . The edition of the A1 being used here (footnote 14), has italicized the ‘theorems’ (here the paragraphs beginning ‘If a connection …’) which are also set off as separate paragraphs, neither of which was done in the original 1844 edition. This is one of the consistent differences which makes the Werke edition easier to read than the 1844 edition and which must be taken into account in using the more recent version for the text of the original. The italicizing and the use of subheadings (though from an outline by Grassmann, these also are unique to the Werke edition) will be ignored in quoting from the A1.
- In Grassmann's H. Lehrbuch der Mathematik für höhere Lehranstalten Erster Theil: Lehrbuch der Arithmetik für höhere Lehranstalten Enslin Berlin 1861 most theorems are given in symbolic form and also stated in words. Since the expression in words appears to play a logical role in the Lehrbuch (taking the place of a rule of substitution) it may be that Grassmann never thought of the use of special mathematical symbols as more than abbreviations for verbal expressions.
- Commentators have assumed that Grassmann had available the terms from the Latin for ‘associative’ ‘commutative’ and ‘distributive’ but chose to make use of German words. For further discussion of Grassmann's terminology see Werke 3 259 259 pt. 2 and the editorial note in vol. 1, pt. 1, 406; and on the tendency of his brother Robert to germanicize foreign terms, see J. E. Hofmann's Introduction to R. Grassmann (footnote 91).
- A1 35 – 36 .
- A1 36 – 36 .
- A1 36 – 36 .
- A1 37 – 38 .
- A1 38 – 39 .
- Cassina , U. , ed. 1958 . Opere scelte di Giuseppe Peano , Vol. 2 , 366 – 366 . Rome : Cremonese .
- A1 39 – 40 .
- A1 40 – 41 .
- Kennedy , H.C. 1974 . Peano's concept of number . Historia mathematica , 1 : 387 – 408 . (p. 404)
- Review of Natucci A. Il concetto di numero, e le sue estensioni, in Areheion, archivo di storia della scienza 1923 4 382 383
- A1 41 – 41 .
- Werke , 1 407 – 407 . pt. 1
- A1 41 – 42 .
- When Grassmann tries to give a more detailed and explicit account of such relations between connections of different steps (addition-subtraction, multiplication-division, and power-root) in his Lehrbuch (footnote 133) it is without much greater success. In his Lehrbuch der Arithmetik und Algebra für Lehrer und Studirende Teubner Leipzig 1873 274 274 E. Schröder, who follows out the program of Grassmann's Lehrbuch to a great extent, refers to the rule in Grassmann's Lehrbuch, p. 313, for changing second-step theorems to third-step theorems as too general ‘to the extent I understand it correctly’, and implies it could result in false theorems.
- A1 43 – 43 .
- This assumption is made explicit in the corresponding proof in the Lehrbuch Enslin Berlin 1861 Theorem 143
- A1 44 – 45 .
- A1 47 – 47 .
- A1 49 – 49 .
- A1 49 – 49 .
- A1 50 – 50 .
- A1 51 – 51 .
- A1 51 – 51 .
- A1 52 – 52 .
- A1 53 – 54 .
- A1 55 – 55 .
- A1 56 – 56 .
- A1 59 – 59 .
- 1839 . “ Ableitung der Krystallgestalten aus dem allgemeinen Gesetz der Krystallbildung ” . In Programm der Ottoschule Stettin reprinted in Werke, vol. 3, pt. 2, 115–146. This is essentially a rewriting of his father's work (footnote 86)
- A1 62 – 62 .
- If the number thus produced [by counting] is made the basis for a new counting by putting it in place of the unit, then the arithmetic contact to multiplication is obtained which is therefore nothing other than a number of higher step, a number whose unit is also a number’ Grassmann J. Raumlehre für die untern Klassen der Gymnasien, und für Volkschulen. Zweiter Theil. Ebene räumliche Grössenlehre G. Reimer Berlin 1824 195 195
- A1 77 – 77 .
- A1 77 – 78 .
- A1 79 – 79 .
- A1 56 – 56 .
- Grassmann , J. 1829 . Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft , 9 – 9 . Stettin : Morin .
- Palter , R. 1972 . Kant's formulations of the laws of motion . Synthèse , 24 : 96 – 116 . in Part II of explains the philosophic importance of Kant's ‘proof’ of the parallelogram rule.
- Bruce , R.V. 1973 . Bell: Alexander Graham Bell and the conquest of solitude , 251 – 252 . Boston : Little Brown .
- A1 79 – 80 .
- A1 80 – 81 .
- A1 81 – 81 .
- A1 82 – 82 .
- A1 82 – 83 .
- A1 108 – 108 .
- A1 108 – 108 .
- A1 84 – 84 .
- A1 84 – 84 .
- A1 207 – 207 .
- A2 ( = Werke Vol. 1 , 10 – 10 . pt. 2
Annals of Science
Volume 34, 1977 - Issue 2
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