Herman Günther Grassmann

This category will include items by or about Robert (brother) and Justus (father) as well as Hermann Grassmann as listed below with post and citation prefixes for each work referenced. Also for Friederich Schleirmacher.

Focus here will be on philosophy and presentation. Actual mathematics using modern notation (including terminology) in separate category for “Universal Geometric Calculus”.


Extension Theory” (Ausdehnungslehre 1862) Tr by Lloyd C Kannenberg.

(History of Mathematics 19) Hermann Grassmann-Extension Theory-American Mathematical Society, London Mathematical Society (2000).pdf

title = {Extension Theory},
author = {Hermann Grassmann},
publisher = {American Mathematical Society, London Mathematical Society},
isbn = {0821820311,9780821820315},
year = {2000},
series = {History of Mathematics 19},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=f9e2162106d2f0aac9ca3c5eff8bf738}

DOI 10.1090/hmath/019 at AMS not accessible via Sci-Hub but is in Library Gensisis.

Freely available A2 endmatter has 107pp including Translator’s Note, Table of Contents, Preface, Foreward, End Notes and subject index:

A2N, A2P, A2E followed by original (not pdf) page number for Translator’s Note, Preface and End Notes (with any numbered paragraph before page number).

Searched Library Genesis.

Got all (English) titles in above Library Genesis search by Hans Joachim Petsche, plus Peano, Collins and Hyde.

title = {Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar: Papers from a Sesquicentennial Conference},
author = {Albrecht Beutelspacher (auth.), Gert Schubring (eds.)},
publisher = {Springer Netherlands},
isbn = {978-90-481-4758-8,978-94-015-8753-2},
year = {1996},
series = {Boston Studies in the Philosophy of Science 187},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=099ef68a7f19595aef3b5759a0ab653b}

title = {From Past to Future: Graßmann’s Work in Context: Graßmann Bicentennial Conference, September 2009},
author = {Hans-Joachim Petsche, Albert C. Lewis, Jörg Liesen, Steve Russ (auth.), Hans-Joachim Petsche, Albert C. Lewis, Jörg Liesen, Steve Russ (eds.)},
publisher = {Birkhäuser Basel},
isbn = {3034604041,9783034604048},
year = {2011},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=96895BE1F2D675503968E494879E85AF}

title = {Hermann Grassmann: Biography},
author = {Hans-Joachim Petsche},
publisher = {Birkhäuser Basel},
isbn = {3764388595,9783764388591},
year = {2009},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=3C36C29161D17FB08C5DC23B0F9C361D}

title = {Hermann Graßmann Roots and Traces: Autographs and Unknown Documents},
author = {Hans-Joachim Petsche, Gottfried Keßler, Lloyd Kannenberg, Jolanta Liskowacka (eds.)},
publisher = {Birkhäuser Basel},
isbn = {978-3-0346-0154-2,978-3-0346-0155-9},
year = {2009},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=865cf5c4e2c9d1eb239772256a806526}



title = {Geometric Calculus: According to the Ausdehnungslehre of H. Grassmann},

author = {Giuseppe Peano (auth.)},

publisher = {Birkhäuser Basel},

isbn = {978-1-4612-7427-8,978-1-4612-2132-6},

year = {2000},

series = {},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=310270CD2C94EA598B18B27729A95A31}}



title = {An elementary exposition of Grassmann’s Ausdehnungslehre, or Theory of extension},

author = {Collins J.S.},

publisher = {Am.Math.Mon.},

isbn = {},

year = {1900},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=42D5156C4DFD533FC646A8C9D5554888}}



title = {The directional calculus, based upon the method of Grassmann},

author = {Hyde E.W.},

publisher = {Ginn},

isbn = {},

year = {1890},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=037EB7F6E6F1BDBC5740B9C957562BBF}}



title = {Grassmann’s space analysis},

author = {E. W. Hyde},

publisher = {Wiley},

isbn = {9781429702256,1429702257},

year = {1906},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=9044E816ABB995DFBD548EE194C4EEB0}}

Want philosophical introduction but:

Translator’s Note

GRASSMANN’S Ausdehnungslehre of 1862 is his most mature presentation of his system, and is unique in capturing the full sweep of his mathematical achievement .

Stripped of the philosophical drapery of his earlier Lineale Ausdehnungslehre of 1844, the “Second Ausdehnungslehre” also contains an enormous amount of material not included (or only suggested) in the earlier book (which, as its title suggests, was deliberately restricted to the “lineale ” aspects of the theory, …


So also want English translation of 1844 “Lineale Ausdehnungslehre also translated by Lloyd C Kannenberg


title = {A New Branch of Mathematics: The Ausdehnungslehre of 1844, and Ohter Works},
author = {Hermann Grassmann},
publisher = {Open Court Pub Co},
isbn = {0-8126-9275-6,0-8126-9276-4,9780812692754},
year = {1996},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=cff320784aa3346c9e25df7ec868ea45}

nLAB has a page on the 1844 A1 edition including pdf scan (German) and quotes from Bill Lawvere:

Discussion of the book includes

William Lawvere, Grassmann’s Dialectics and Category Theory, in Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264 (publisher)

and the similar text

William Lawvere, A new branch of mathematics, “The Ausdehnungslehre of 1844,” and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis, Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106 (publisher)

which says at one point that full appreciation of the Ausdehnungslehre requires concepts of category theory

The modern conceptual apparatus, involving levels of structure, categories of morphisms preserving given structure, forgetful reduct functors between categories, the adjoints to such functors, etc., seems to be necessary for ordinary mortals to be able to find their way through the riches of Grassmann’s geometry.

The first part of the introduction of the Ausdehnungslehre is concerned with philosophy, about which Grassmann insists that his reason for including it is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process (Lawvere 95).

The second part of the introduction, titled Survey of the general theory of forms considers key concepts of algebra. For instance it considers the associativity law and states its coherence law (§3). Grassmann writes that he uses the term “form” in place of “quantity” (German: “Grösse”) (Introduction A.3, §2). It is “forms” that his algebraic operations are defined on, and which are produced by these.

The last half of that introduction is essentially one of the first expositions of the rudimentary principles of what today might be called universal algebra. The content of the first half, after considerable study of the compact formulations, appears to be a simple and clear natural scientist’s version of the basic principles of dialectical materialism, as applied to the formal sciences. (Lawvere 95)

Curiously, while Grassmann complains (on p. xv) about the “unclarity and arbitrariness” of Hegel’s school of philosophy (German idealism, predominant in Germany at Grassmann’s time), the introduction of the Ausdehnungslehre has much the same sound as Hegel, notably it discusses “categories” such as being, becoming (p. xxii), particulars (p.xx) and the dialectic of opposites such as discrete ⊣\dashv continuous (p.xxii) and, notably, of intensive and extensive quantity (p. xxiv-xxv), which Grassmann advertizes as the very topic of his mathematical theory. That of course is the difference to Hegel, that unambiguous mathematical formalization of these otherwise vague concepts is provided (according to Lawvere 95 Grassmannn’s formalization of the pair being and becoming is via points and vectors in an affine space), and in this sense Grassmann is clearly a forerunner of Lawvere’s various proposals for formalizing Hegel’s objective logic in categorical logic/topos theory (as discussed at Science of Logic).

1. References

Wikipedia, Grassmann – Mathematician


Bill Lawvere’s detailed 7pp review of the translations of both 1844 and 1862 editions including above emphasis added remark on “basic principles of dialectical materialism” is here at doi https://doi.org/10.1016/j.hm.2004.07.004 (pdf)


Full version is Lawvere F.W. (1996) Grassmann’s Dialectics and Category Theory. In: Schubring G. (eds) Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Boston Studies in the Philosophy of Science, vol 187. Springer, Dordrecht. p255-264

DOI https://doi.org/10.1007/978-94-015-8753-2_21 ISBN 978-90-481-4758-8 Online ISBN 978-94-015-8753-2



title = {Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar: Papers from a Sesquicentennial Conference},

author = {Albrecht Beutelspacher (auth.), Gert Schubring (eds.)},

publisher = {Springer Netherlands},

isbn = {978-90-481-4758-8,978-94-015-8753-2},

year = {1996},

series = {Boston Studies in the Philosophy of Science 187},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=099EF68A7F19595AEF3B5759A0AB653B}}

Got both above. Lawvere VERY interesting but still above my level. Schubring sesquicentennial also looks interesting but not read yet.

Crowe History of Vector Analysis also has detailed material, chapter 3.

Other copies poor or missing pages. Best copy is:

Michael J. Crowe-A history of vector analysis_ the evolution of the idea of a vectorial system-Dover (1985).djvu


title = {A history of vector analysis: the evolution of the idea of a vectorial system},
author = {Michael J. Crowe},
publisher = {Dover},
isbn = {9780486649559,0486649555},
year = {1985},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=FF4DAC18A376DE02AE307713C7885077}


Albert C. Lewis (1977) H. Grassmann’s 1844 Ausdehnungslehre
and Schleiermacher’s Dialektik , Annals of Science, 34:2, 103-162, DOI:

@article{Lewis-1977,@article{Lewis-1977,doi = {10.1080/00033797700200171},title = {H. Grassmann’s 1844 Ausdehnungslehre and Schleiermacher’s Dialektik},author = {Lewis, Albert C.},publisher = {Taylor and Francis Group},journal = {Annals of Science},issnp = {0003-3790},issne = {1464-505X},year = {1977},month = {03},volume = {34},issue = {2},page = {103–162},url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1080/00033797700200171},}

LGSD has extensive quotation from philosophical side and useful explanations. But note Lawvere’s remarks on religious propaganda from American Mathematical Society imposed by patron of Paul Carus “The Monist”.

Schleiermacher Dialectic earlier 1811 version only – english:


Found at archive.org but turns out to be German. Only seen a chapter on dialectic in book on Schleirmacher’s heremeneutics (of which he is regarded as founder).


From Past to Future: Grassmann’s Work in Context

Grassmann Bicentennial Conference
(1809 – 1877)
September 16 – 19, 2009 Potsdam / Szczecin (DE / PL)


Has works (in German) by both Hermann and brother Robert. Note sequicentennial above also has (english translation of) important paper by father Justus and lots of other useful stuff.

Will post comments on Kannenberg’s translation “A New Branch of Mathematics” below with following prefixes so that alphabetic sort is in page order but with the editorial end notes for each work included with the relevant portions by suffix “N” and note number, footnotes “n” and number:

Front Matter (with two digit page numbers replacing roman numerals i-xvi):

A1Bf – Foreward by Albert C Lewis (09-10)

A1Bs – Selected Sources (10-11)

A1Bt – Translator’s Note by Lloyd C. Kannenberg (13-16)


A1E – Linear Extension Theory (1878 edition of the 1844 original) (p1-278)

A1Ef1. – Foreword to the first edition – 28 June 1844 (09-17)

A1Ef2. -Foreword to the second edition – Summer 1877 (19-22)

A1Ei – Introduction (23-32) paragraph numbers 01-16 will be used instead of A to D below.

A1Ei 01-03 A – Derivation of the Concept of Pure Mathematics (23-25)

A1Ei 04-08 B – Derivation of the Concept of Extension Theory (25-28)

A1Ei 09-12 C – Exposition of the Concept of Extension Theory (28-30)

A1Ei 13-16 D – Form of Presentation (30-32)

A1Ess – paragraph numbers 001-172 as listed in Table of Contents for Part 1, pp3-8

A1EA1 – Appendix I (1877) – On the relation of Non-Euclidean Geometry to Extension Theory (p279-280)

A1EA2 – Appendix II (1877) – On the Regressive Product (p281-282)

A1EA3 – Appendix III (1877) – Brief Survey of the Essentials of Extension Theory – numbered paragraphs 01 to 22, followed by P and independently numbered problem number 01-23 where the reference is to a problem rather than the preceding numbered paragraph.

A1EI – Index of Defined Terms (1877) p296-297

Editorial Notes for Part I, 01-54 on pp298-312 are suffixed as Nnn following the above prefixes to sort together with the references instead of at the end.


A1Gss Geometric Analysis (1847) pp313-384 as numbered paragraphs from 00 (Introduction) to 24 (Conclusion)

A1Mss The Grassmannian Theory of Point Magnitudes and the Magnitude Forms Depending on Them by A.F Mobius as numbered paragraphs from 00 to 19.

Editorial Notes for Part II on pp403 to 414 are suffixed as N00 to N13 followed by page number where useful (ie for N00p403-409).

Part III

A1P Selected Papers on Mathematics and Physics pp415-542

A1Z backmatter is pp543-545 Bibliography (principal works in German) and pp547-555 Index.

title={Lehrbuch der Arithmetik f{\”u}r h{\”o}here Lehranstalten},
author={Grassmann, H.},
series={Lehrbuch der Matematik f{\”u}r h{\”o}here lehranstalten ; 1. Th},
publisher={Th. Chr. Fr. Enslin}

Unpublished (2005) translation by Lloyd Kannenberg of Robert Grassmann “Laws of Form” (Formenlehre) arrived (10 days from email 2017-12-08)

“A couple of further points: 1. This is a translation of the 1872 edition of Die Formelehre oder Mathematik, not the 1895 Die Formenlehre oder Mathematik in strenger Formentwicklung. 2. I understand that a reprint of Hermann Grassmann’s Lehrbuch der Arithmetik, the first part of his Lehrbuch der Mathematik mentioned in the Translator’s Note, is now available.”

Original 1872 in German download from:

Die Formenlehre oder Mathematik : [= Zweiter Ergänzungstheil]
Autor / Hrsg.: Grassmann, Robert ; Grassmann, Robert
Verlagsort: Stettin | Erscheinungsjahr: 1872 | Verlag: Grassmann
Signatur: 12597345 Ph.u. 224 p-6,1/3 12597345 Ph.u. 224 p-6,1/3
Reihe: Die Formenlehre oder Mathematik : [= Zweiter Ergänzungstheil]
Permalink: http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb11018135-6


Original 1895 in German download from:


See also:


Assume that mentioned first part of Hermann Grassmann “Lehrbuch der Arithmetik” refers to 1861 German original, not translation:


Google Scholar has 101 related articles for “Lehrbuch der Mathematik”.


Also 78 citations (all to “Lehrbuch der Arithmetik”):


Hao Wang on “The Axiomatization of Arithmetic” very useful exposition of Dedekind/Peano/Grassmann arithmetic


@article{Ha-1957,@article{Ha-1957,doi = {10.2307/2964176},title = {The Axiomatization of Arithmetic},author = {Hao Wang},publisher = {Association for Symbolic Logic},journal = {Journal of Symbolic Logic},issnp = {0022-4812},issne = {1943-5886},year = {1957},month = {06},volume = {22},issue = {2},page = {145–158},url = {http://gen.lib.rus.ec/scimag/index.php?s=10.2307/2964176},}

Very important. Describes arithmetic from Hermann’s Lehbuch of 1861. Looks like same approach as Robert 1872 (and done in collaboration). Says Peano used it after getting from Dedekind 1888 (preceded by Frege 1884).

Dedekind uses term “Anzahl” (cardinality) – see Klein re important distinction.

Hao Wang discusses weaker forms of induction and draws attention to Peano/Dedekind reliance on set theory, but seems to miss what I think is essential. Both Grassmanns used strictly constructive “generation” that is a purely equational calculus (with no existence predicate) and does NOT require any such infinite metaphysics.

Starts from a basic induction on formulas that looks to me much more like Goodstein than Peano. See refs in finitism.

Very much in the constructive tradition from Euclid consistent with their education. Should it be regarded as a precursor to modern foundations based on Constructive Higher Order Logic (and modal homotopy type theory).

Michael Potter does Dedekind in modern notation (closures for “chains”).

title={Set Theory and its Philosophy: A Critical Introduction},
author={Potter, M.},
publisher={Clarendon Press}

(Got hardcopy)

Could be nice to derive “philosophically” like Grassmann from absolute minimum (similar/different – both irreflexive) in terms of Topos with a “number object”. (Grassmann already uses concepts like composition of injective maps being injective, and endomaps being closures). “Forms” essentially predicates conjoined intensionally not extensionally? Extensions could be properties/classes, binary relations, maps or difunctions (semi-single valued).

Robert seems to have come close to concrete relational algebras. Not finished reading “binding” (combinations) or logic of concepts (monadic predicate calculus). But looks as though should have noticed at least that “simple relation” for symmetric difference is union. Does boolean algebra on powerset of direct product with relational composition fit with theory of forms?

More english language Grassmann or Extension theory related in CHV refs especially:

Book IV “Theory of Extension Theory” TUA4 in:


title = {A Treatise on Universal Algebra: With Applications},
author = {Alfred North Whitehead},
publisher = {Cambridge University Press},
isbn = {1108001688,9781108001687},
year = {2009},
series = {Cambridge Library Collection – Mathematics},
edition = {Reissue},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=8c87bdb9f9f0a244dbdca42279b7cae1}


title = {The Calculus of Extension },
author = {Henry George Forder, Robert William Genese},
publisher = {The University press},
isbn = {},
year = {1941},
series = {},
edition = {1st},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=8D9DCDD31AA96B908EE863E24CB865E6}}


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