Hermann Graßmann: Philologist and Mathematician

Today marks the two-hundred-and-third birthday of the German linguist and polymath Hermann Grassmann (Graßmann, 15.04.1809–09.26.1877).  Hermann Grassmann is, perhaps, not well known within Classical Studies, although anyone who has studied Ancient Greek in any depth would likely have come across his name at some point.  In 1863 Hermann Grassmann discovered a rule of Greek and Sanskrit phonology which predicted irregularities in certain nominal and verbal derivations from lexical roots containing aspirated stop consonants (in Greek: φ [pʰ], θ [tʰ], and χ [kʰ]).[1]
The rule, as Grassmann originally propounded it, was as follows:

(1) ‘Given a root with a final aspirate and an initial consonant capable of aspiration, and given also that the final element loses aspiration (by some separate sound law), then that feature is retracted to the initial element.’

(2) ‘Given two consonant groups in a word, separated by a vowel and themselves aspirated, and provided that they are within the same root, then one (and normally the first) is deprived of its breath feature.’[2]

The first part of this rule, is actually only applicable to certain phonetic environments in Sanskrit, but the second part is very much the case where you find, for example, synchronic alternations in the stem of the word θρίξ ‘hair’, which has an underlying root *ΘΡΙΧ- [tʰrikʰ-] exhibits the following paradigm (with morpheme boundaries indicated):

N θρίξ [tʰrík-s]
G τριχός [trikʰ-ós]
D τριχί [trikʰ-í]
A τρίχα [tríkʰ-a]


N τριχές [trikʰ-és]
G τριχῶν [trikʰ-ō̃n]
D θριξί [tʰrik-sí]
A τρίχας [tríkʰ-as]

Also the same dissimilation is easily seen in the notoriously irregular principal parts of the verb ἔχω [ekʰ-ō] ‘I have, hold’ (the root is *seg̑ʰ- cf. Germ. Sieg, and word initial /s/ becomes /h/ before a vowel in prehistoric Greek), which has two competing irregular future forms ἔξω [hek-s-ō] (with root e-vocalism) and σχήσω [s(ø)kʰ-ēs-ō] (with root-zero vocalism), and an aorist stem ἔσχον  [e-s(ø)kʰ-o-n], (also with zero root vocalism).  Also, perhaps a little more transparently, in the stem formations of reduplicated μι-conjugation verbs τίθημι ‘I place’ (< *tʰi-tʰē-mi) and  ἵστημι ‘I stand’ (< *si-stā-mi), in comparison to those which do not have an aspirated root initial consonant, like δίδωμι ‘I give’ (< *di-dō-mi).

As such, this law, Grassmann’s Law as it has come to be known, explains much of the irregularities in lexical roots with aspirates as they surface in Greek and Sanskrit, and although the phenomenon arose independently in Greek and Sanskrit, as the example from Gothic bindan ‘bind’ is cognate to Sanskrit √bandh- ‘bind’, and Young Avestan baṇdaiieiti ‘id.’  A complicated argument can be made that Grassmann’s Law was not in operation in the Mycenaean Greek language of the second-millennium Linear B archives, that would a require lengthy and off-topic digression.

Although this sound law was also applicable to Greek as well as Sanskrit, Grassmann’s main interests, however, were mainly with Vedic studies.  One of his principal publications in that area effort, his 1872 Wörterbuch zum Rig-Veda (Dictionary of the Rig-Veda), remains for the most part unsuperceded, and a translation of the Rig-Veda itself into German.

It is also worthwhile to note about Grassmann, that while he was respected in his own lifetime as a general linguist and an important contributor to Vedic and Indo-European philology, it is little known among philologists that prior to coming to these studies Grassmann’s first, and perhaps most interesting work was actually in mathematics.  His 1844 work Die lineale Ausdehnungslehre: Ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], effectively single-handedly founded Linear Algebra.  This work was submitted as a Ph.D. thesis in Mathematics, however its formulation of linear vector space in opposition to the canonical Euclidean geometry of the time was too radical for his contemporary mathematics establishment and was rejected.  Consequently, the significance of contribution to the mathematical sciences were largely unrecognized in his lifetime, but they were eventually re-discovered towards the end of the nineteenth century and early twentieth century.  It was because of these rejections and early lack of recognition of his work in the mathematics that he suffered he turned himself to Vedic studies, and made those discoveries in there which we know him best for, and for which Grassmann did, eventually, earn an honorary doctorate of letters from the University of Tübingen in 1876.

Grassmann was also the author of another Grassmann’s Law, the ‘first’ Grassmann’s Law of 1853 which is in still taught in optics, regarding the mixture of different light colours in linear combination.
[1] Grassmann, H. 1863. “Über die Aspiration und ihr gleichzeitiges Vorhandensein in an- und auslaute der Wurzeln.” Zeitschrift für vergleichende Sprachforschung auf dem Gebiete des Deutschen, Griechischen und Lateinischen 12.81–138 (Available at http://www.jstor.org/stable/40844346)

[2] Collinge, N.E. 1985. “Grassmann’s Law” In: id. The Laws of Indo-European. Amsterdam: John Benjamins. 47–61

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About Matt Scarborough

न वदेद्यावनीं भाषां प्राणैः कण्ठगतैरपि । «One should not speak a Western language even to save one's life.»
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3 Responses to Hermann Graßmann: Philologist and Mathematician

  1. Pingback: Hermann Graßmann at 203 « Memiyawanzi

  2. johnwcowan says:

    Who knew that all these Grassmanns were the same fellow?

    Similarly, I didn’t know until recently that Lomosonov the biologist and Lomosonov the poet and grammarian (of Russian) were one and the same….

  3. Pingback: Two Years of Res Gerendae | res gerendae

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