© Springer Nature Switzerland AG 2020
R. Shamey, R. G. KuehniPioneers of Color Sciencehttps://doi.org/10.1007/978-3-319-30811-1_38

38. Grassmann, Hermann Günther 1809–1877

Renzo Shamey1   and Rolf G. Kuehni1  
(1)
Color Science and Imaging Laboratory, North Carolina State University, Wilson College of Textiles, Raleigh, NC, USA
 
 
Renzo Shamey (Corresponding author)
Email: rshamey@ncsu.edu
 
Rolf G. Kuehni
Email: rolf_kuehni@ncsu.edu
38.1 Grassmann’s Laws
References
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Grassmann portrait, artist unknown

Hermann Gunther Grassmann was born on April 15, 1809, in Stettin in Pomerania, near the Baltic Sea (today Szczecin in Poland), the third of 11 children of a pastor and high school mathematics teacher and his wife. After passing through high school, Grassmann moved to Berlin to study theology, with later addition of mathematics and sciences. In 1844, he published his major work in mathematics, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (Linear extension theory, a new branch of mathematics) [1], a general calculus of vectors. Ausdehnungslehre was not a success, apparently clearly ahead of its time. In 1846, Grassmann received an award for expanding on a mathematical problem sketched earlier by Leibniz. Grassmann married in 1849, and he and his wife had 11 children. His father, though teaching at a high school, had been named professor a few years before he passed away in 1852. In that same year, Hermann Grassmann assumed the position of mathematics professor his father had held at the Stettin Gymnasium. In his later years, unhappy about the continuing lack of attention to his mathematical efforts, he became interested in the history of languages. He learned Sanskrit and prepared a dictionary and a translation of the sacred collection of Indian Vedic hymns, the Rigveda, one of the oldest extant written records in an Indo-European language, dating to the mid-second millennium BCE [2]. Both works immediately gained much admiration and support from linguists. Grassmann died on September 26, 1877, in Stettin.

38.1 Grassmann’s Laws

In 1852, Hermann von Helmholtz published an article, based on his results of experimental work in color mixture with a spectroscope of his own design. He concluded that Newton’s structural design of his color circle, based on his own experiments with mixing spectral lights, must be in error and that there are only two spectral colors, blue and yellow, that when mixed result in colorless appearance [3]. Grassmann applied mathematical logic to the problem and in 1853 published a paper “Zur Theorie der Farbenmischung” (On the theory of color mixture) [4] in the same journal as Helmholtz, claiming that Helmholtz was likely in error. He postulated four “assumptions” about color mixture:
  1. 1.

    Every impression of color may be analyzed into three mathematically determinable elements—hue, intensity of color, and brightness of the intermixed white light.

     
  2. 2.

    If one of two mingling lights is continuously altered (while the other remains unchanged), the impression of the mixed light is also continuously changed.

     
  3. 3.

    Two colors, both of which have the same hue and the same proportion of intermixed white, also give identical mixed colors, no matter what homogeneous colors they may be composed of.

     
  4. 4.

    The total intensity of any mixture is the sum of the intensities of the lights mixed.

     
A modern interpretation of the content of assumptions 2–4, as provided by Wyszecki and Stiles, [5], are the following four laws:
  1. 1.

    Symmetry law: If color stimulus A matches stimulus B, then stimulus B matches stimulus A.

     
  2. 2.

    Transitivity law: If A matches B and B matches C, then A matches C.

     
  3. 3.

    Proportionality law: If A matches B, then aA matches aB, where a is a positive factor of the radiant power of the stimulus.

     
  4. 4.

    Additivity law: If A matches B and C matches D, then (A + D) matches (B + C) (applicable to additive mixtures).

     

These laws do not explicitly consider variations in conditions, in eye adaptation, or variation in color matching functions between observers (see, e.g., Brill and Robertson) [6].

The laws are considered fundamental components of a trichromatic theory of color vision. In response to Grassmann’s paper, after modifying his spectroscope, Helmholtz redid the color mixing experiments and was able to determine multiple complementary pairs in the spectrum, thus confirming the basic validity of Grassmann’s laws [7].