Thus, let s (fig. 1) be a very slender pencil of rays of some one colour incident on a spherical drop of water at the angle A r s, and let this angle be such that the rays in the pencil may, by the laws of refraction in water, converge at B; then, though many rays will pass through the drop at that point and be dispersed, yet many will be reflected from thence as from a radiant point, and will emerge at K in parallel directions, as they entered at r, so that if x x be the direction of the emergent pencil, the angle o x x will be equal to A is : the angle made by the lines a r and E R produced was found by Descartes to be about 42 degrees. If the angle A i s were varied, the rays of the pencil would leave the drop in a divergent state, and then the impression which they would make on the eye might be too feeble to produce the sensation of brightness. Again, let s i (fig. 2) be a very slender pencil of rays of some one colour incident on a spherical drop of water at the angle A i s, and let this angle be such that, by the laws of refraction in water, the rays, after crossing at x and being reflected from n, may pass from B to c in parallel directions; then, after a second reflection crossing at Y and being refracted at fc, they will emerge in pamlle directions as they entered at 1, so that if K E be the direction of the emergent pencil, time angle n will be equal to A 1 5 : the angle made by the lines 81 and E K was found by Descartes to be about 52 degrees. If the angle A r s were varied, the rays of the pencil would leave the drop in a divergent state.
Now let A, a, D (fig. 3), be four globules of rain in a cloud covering are reflected towards the upper surface, and there they suffer a second reflection. After this they pass to the side of the drop which is nearest to the sun, and from thence they emerge after a second refraction.
a considerable part of the heavens on one side of the horizon. Let E be the eye of the spectator, and, on account of the remoteness of the sun, let the rays of light which proceed from his disc be considered AS parallel to one another. Let s E be a line drawn from the sun through the eyo of the spectator, and let it be produced towards o; also let s s n, fto., bo very slender pencils of parallel raya (supposed at present to be of one colour) falling upon the globules of water. Let the refraction and reflection of these pencils in A and u be similar to those which aro shown in fig. 1 ; and the refraction and reflection In o and n be similar to those iu fig. 2 ; also from the points of emergence suppose lines to be drawn to s. It is evident, on account of the parallelism cf the lines s 0, 5 A, ft:c., that if the angle A 13 0 or B 13 0 were nearly equal to 42°, and if the angles o E o or DE o were nearly equal to 52°, the eye would be affected by the sensation of brightness as explained above ; therefore, if the lines a e, BE, &c., were to revolve conically about E o as an axis,
all the globules of rain upon the conical surfaces so described would send pencils of parallel rays to the eye, and two concentric arches of bright light would be seen in the heavens. This hypothesis accounts satisfactorily for the existence of two concentric bows of bright light, but it affords no indication of the bands of colours of which they con sist. Descartes, however, very sagaciously refers their cause to the decomposition of light on entering and quitting the drops of rain, observing that the convex surfaces of the drops must produce effects similar to those which take place when light is made to pass through the plane faces of a triangular prism of water.
But when Newton bad discovered the different degrees of refrangi bility in the different coloured rays which composed a pencil of white or compounded light, he was able to assign immediately the cause of the coloured bands in the rainbow, the order of their position, and the breadth which they must occupy. Thus, if the incident pencil (Jigs. 1 and 2) had consisted only of violet-coloured light (for example), the angle A I s must have had that particular value which alone would allow the rays of the emergent pencil to be parallel to one another ; but if the incident pencil were supposed to consist of light of another colour, as red, it should have fallen a little further from the centre of the drop, in order that the angle A I a might have the particular value which would allow the rays of the emergent pencil to be parallel to one another. It may be proved without difficulty that the total deviation, when consecutive emergent rays of a given colour come out parallel to one another, is less for red rays than for violet ; and if Ks be the direction in which the latter emerge from a drop, IC I in both figures may represent the direction in which the former would emerge, the variation of the course before emergence being omitted in the figures to avoid confusion ; and if the eye were situated so as to receive the pencil K 5, it would have the impression of a violet colour ; while, if it were situated so as to receive the pencil s, it would have that of a red colour. We have mentioned, for sim plicity, only the violet and red rays, which form the two extremes of the coloured spectrum ; but it is easy to conceive that a like explana tion might be given for rays of the intermediate colours. And since the pencils of all the different colours diverge from one another on quitting a rain-drop, it is evident that the spectator whose eye receives one of the pencils will be affected by the colour of that pencil only, the other pencils passing either above or below his eye.