# From G. H. Darwin   10 October 1878

Trin Coll. Camb.

Oct 10. 78

My dear Father,

The only experiments on Fechner’s law wh. I have found in Helmholz are experiments on the smallest differences of illumination which are visible.1 He F. found that the smallest perceptible difference was always the same fraction of the intensity. He put two candles before a screen & held a rod in front which cast two shadows. He then withdrew one candle further & further until one could no longer see the shadow cast by it; the other candle being kept still. This was performed a number of times with the fixed candle at various distances from the screen   He always found that when the moveable candle was about 10 times as far from the screen as the fixed one that the shadow cast by the moveable one was just imperceptible. The illumination due to the moveable one was of course 10 squared or 100 times as small as that of the fixed one.— That is to say a difference of $\frac{1}{100}$th of the total illumination was just imperceptibly— (The number of course varied for different eyes). From this may be deduced the law as it is usually stated with logarithms but it seems much more intelligible as it stands.2

It is expressly stated that it was no longer exact for very bright lights because of the fatigue or straining of the eyes—

Nor for very weak lights probably because of the innate light of the eyes—(which produces that sort “luminous chaos” when you shut your eyes for long eno’). According to this you might make some sort of attempt to prove Fechner’s law by seeing what was the least difference of distance from the lamp which made any perceptible differences in the bending of the seedlings.3

Suppose that there was just a perceptible difference when a pot was 2 ft & 2 ft 6. from the light Then the difference of intensities wd be

$\frac{1}{2}$2-1/(2$\frac{1}{2}$)2=$\frac{1}{4}$-$\frac{4}{25}$=$\frac{9}{100}$

& the intensity of the light in the first position was

$\frac{1}{2}$2=$\frac{1}{4}$

Now $\frac{9}{100}$ is $\frac{36}{100}$ of $\frac{1}{4}$

Therefore this experiment would assert that the seedling could perceive a change of $\frac{36}{100}$ of the light which fell on it.

Then at 6 feet the light is $\frac{1}{36}$ & if two plants at 6 feet & 7 ft 6 could just perceive the difference Fechner’s law wd. be true—for

$\frac{1}{6}$2-1/(7$\frac{1}{2}$)2 when divided by $\frac{1}{6}$2 is $\frac{36}{100}$ as in the first supposed experiment.

If at any distance from the lamp you can give the greatest amount of change in the position of a pot which just makes no difference—(or just makes a difference which is I suppose the same thing)—then one can give a series of numbers which according to Fechner’s Law ought to express the magnitude of the effects for various stimulants.4 Without such a datum it is not possible. Accordingly it is impossible to give a series of numbers which you could compare with your experiments.— at least so it seems to me.

I’m rather seedy this afternoon & can’t write properly.

Yours affec | G H Darwin

## Footnotes

Fechner’s law was first stated in Gustav Theodor Fechner’s work Elemente der Psychophysik (Elements of psychophysics; Fechner 1860, 2: 13). The law was an attempt to quantify the relationship between a physical stimulus and the perception of the stimulus. For Hermann von Helmholtz’s discussion of Fechner’s law in relation to perception of light intensity, see Helmholtz 1867, pp. 309–16.
Fechner, in studies on differences in light intensity, concluded that the perception of the difference was proportional to the logarithm of the actual measured intensity (Fechner 1860, 2: 13; 31–3 et passim).
CD was studying the sensitivity to light of cotyledons in many different plant species (see, for example, Correspondence vol. 27, letter to Francis Darwin, 3 June [1879]).
George refers to the discovery by Ernst Heinrich Weber, formalised by Fechner as Weber’s law (Fechner 1860, 1: 134), that two sensations are just noticeably different as long as the ratio between the strengths in each pair of stimuli remains constant. The constant, which is different for different stimuli, is often referred to now as the ‘just noticeable difference’ or ‘Weber fraction’ (Complete dictionary of scientific biography s.v. Weber, Ernst Heinrich).

## Bibliography

Complete dictionary of scientific biography. By Charles Coulston Gillispie, Frederic Lawrence Holmes, and Noretta Koertge. Electronic publication. Detroit: Charles Scribner’s Sons. 2008.

Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 27 vols to date. Cambridge: Cambridge University Press. 1985–.

Fechner, Gustav Theodor. 1860. Elemente der Psychophysik. 2 vols. Leipzig: Breitkopf and Härtel.

Helmholtz, Hermann von. 1867. Handbuch der physiologischen Optik. Vol. 9 of Allgemeine Encyklopädie der Physik. Leipzig: Leopold Voss.

## Summary

Recounts the experiments on Fechner’s law he has found in Helmholz; they are on the smallest perceptible differences of illumination. Describes how to test whether plants’ responses to lights are in accordance with it.

## Letter details

Letter no.
DCP-LETT-11722
From
George Howard Darwin
To
Charles Robert Darwin
Sent from
Trinity College, Cambridge
Source of text
DAR 210.2: 70
Physical description
8pp