Here I will make two suppositions.
First, That the extreme red rayss at F M, Optics, book I. part 2. fig. 4. excite vibrations in the retina, which are to those excited by the extreme violet rays at A G, as 1 to 2, in respect of frequency.
Secondly, That in going from the extreme red to the extreme violet, the excess of vibrations excited by each colour, above those of the extreme red, will be proportional to its distance from the extreme red.
If we admit these two suppositions, then the vibrations excited by the extreme red, by the limit of red and orange, of orange and yellow, yellow and green, green and blue, blue and indigo, indigo and violet, and by the extreme violet, as these colours are fixed by Sir Isaac Newton, will be to one another in frequency, respectively as the 8 numbers the distances of these several limits, and of the extreme violet, from the extreme red, being to one another respectively, as the 7 numbers
Now the first suppositions may be rendered probable thus. The intervals of the fits of easy reflection and transmission of the red and violet in the same medium, and same angle of refraction, are nearly as 5 to 3. See Optics, book II. Obs. 13, 14, and Prop. XVI. But the red is less refracted by the coats and humours of the eye than the violet, and consequently will not have its intervals so much diminished in proportion; whence they may be to those of the violet as 6 to 3, or 2 to 1, at their arrival on the retina. But it is probable, that the vibrations of the rays themselves, and consequently those which they excite in the retina, are reciprocally as the intervals of their fits. The frequency therefore of the vibrations excited by the extreme red may be to that of the vibrations excited by the extreme violet as 1 to 2, according to the first supposition.
The second supposition is an easy step after the first.
For it is natural to suppose, that in passing from F to A, in the figure above referred to, equal distances should produce an equal increase of vibrations, which is the second supposition.
Upon this foundation we may now reason in the following manner.
First, The seven primary colours, estimated both from their limits, and their middle points, excite vibrations, which are to each other in the simplest ratios that are consistent with each other, and all comprehended within the first and most simple of all ratios, viz. that expressed by the two first numbers 1 and 2.
Secondly, The same ratios are also those of the five tones, and two semi-tones, comprehended within the octave; as might well
