From G. H. Darwin   10 October 1878

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 w^d 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 w^d. 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