1 873.] • Colours and their Relations. 95 



the geometrical progression is regular, all major keys would 

 have been exactly similar, and so would all minor keys. 

 But the departures from regularity make every one major 

 key to differ from every other major key, and so also with 

 the minor, thus affording a much greater variety. 



Now, as regards the correspondence of the scale of 

 colour with one or other of the musical scales, it was at one 

 time thought to be closer than it really is. For the rates 

 of vibration corresponding to the junction of the colours 

 were believed to constitute the following series, — 1, 1*125, 

 1*2, 1*33', i*5, r66', 177', 2, — thus tallying with the har- 

 monic scale in its minor mode. Prof. Listing, however, by 

 a careful comparison of the most recent and accurate ob- 

 servations, — those made by Angstrom and others, — has 

 determined, with a greater approximation to the truth, the 

 wave-lengths corresponding to the borders of the several 

 colours, and has shown that the reciprocals of those wave- 

 lengths, which correspond to the ratios of the vibrations at 

 those points, form a series approaching much more closely 

 to an arithmetical than to a geometrical progression. (See 

 Pog. An., vol. cxxxi., p. 564). 



When the reciprocals of Prof. Listing's wave-lengths 

 have their relations reduced to the simplest form, by making 

 the smallest number _= unity, they form the following 

 series :— 1, 1*117738, 1*235314, 1*352908, 1*470618, 1*588145, 

 1705730, 1-823568. 



There is here an evident approach to a common difference, 

 of which the mean value is 0*117653. 



This approach to an arithmetical progression becomes 

 more apparent when the arithmetical means of the recipro- 

 cals of Listing's numbers are taken. These form the 

 following series : — 



Red. Orange. Yellow. Green. Blue. Indigo. Violet. 



I, I*III064, 1*222108, 1*333213, 1*444288, I-555303, 1*666453. 



It is evident what is the true law of this series, namely, 

 that all of the above numbers should be perfect: repeating 

 decimals, having a common difference of o'n'. . This 

 series, thus corrected, being assumed, it is easy to calculate 

 backwards, so as to show the agreement of this assumption 

 with observation. Taking the green as the central colour, 

 by applying the above corrected series, we obtain from each 

 of the other colours a value of the green ; and the average 

 of these six values will be found to differ by a mere trifle 

 from the value deduced from the observations. From this 

 corrected value of the green all the others may be found by 



