I SHALL now explain the rainbow. The manner of its production was understood, in the general, before Sir Isaac Newton had discovered his theory of colours; but what caused the diversity of colours in it could not then be known, which obliges him to explain this appearance particularly; whom we shall imitate as follows. The first person, who expressly shewed the rainbow to be formed by the reflection of the sun-beams from drops of falling rain, was Antonio de Dominis. But this was afterwards more fully and distinctly explained by DesCartes.2. There appears most frequently two rainbows; both of which are caused by the foresaid reflection of the sun-beams from the drops of falling rain, but are not produced by all the light which falls upon and are reflected from the drops. The inner bow is produced by those rays only which enter the drop, and at their entrance are so refracted as to unite into a point, as it were, upon the farther surface of the drop, as is represented in fig. 160; where the contiguous rays ab, cd, ef, coming from the sun, and therefore to sense parallel, upon their entrance into the drop in the points b, d, f, are so refracted as to meet together in the point g, upon the farther surface of the drop. Now these rays being reflected nearly from the same point of the surface, the angle of incidence of each ray upon the point g being equal to the angle of reflection, the rays will return in the lines gh, gk, gl, in the same manner inclined to each other, as they were before their incidence upon the point g, and will make the same angles with the surface of the drop at the points b, k, l, as at the points b, d, f, after their entrance; and therefore after their emergence out of the drop each ray will be inclined to the surface in the same angle, as when it first entered it; whence the lines bm, kn, lo, in which the rays emerge, must be parallel to each other, as well as the lines ab, cd, ef, in which they were incident. But these emerging rays being parallel will not spread nor diverge from each other in their passage from the drop, and therefore will enter the eye conveniently situated in sufficient plenty to cause a sensation. Whereas all the other rays, whether those nearer the center of the drop, as pq, rs, or those farther off, as tu, wx, will be reflected from other points in the hinder surface of the drop; namely, the ray pq from the point y, rs from z, tv from a, and wx from . And for this reason by their reflection and succeeding refraction they will be scattered after their emergence from the forementioned rays and from each other, and therefore cannot enter the eye placed to receive them copious enough to excite any distinct sensation.3. The external rainbow is formed by two reflections made between the incidence and emergence of the rays; for it is to be noted, that the rays gh, gk, gl, at the points h, k, l, do not wholly pass out of the drop, but are in part reflected back; though the second reflection of these particular rays does not form the outer bow. For this bow is made by those rays, which after their entrance into the drop are by the refraction of it united, before they arrive at the farther surface, at such a distance from it, that when they fall upon that surface, they may be reflected in parallel lines, as is represented in fig. 161; where the rays ab, cd, ef, are collected by the refraction of the drop into the point g, and passing on from thence strike upon the surface of the drop in the points h, k, l, and are thence reflected to m, n, o, passing from h to m, from k to n, and from l to o in parallel lines. For these rays after reflection at m, n, o, will meet again in the point p, at the same distance from these points of reflection m, n, o, as the point g is from the former points of reflection h, k, l. Therefore these rays in passing from p to the surface of the drop will fall upon that surface in the points q, r, s in the same angles, as these rays made with the surface in b, d, f, after refraction. Consequently, when these rays emerge out of the drop into the air, each ray will make with the surface of the drop the same angle, as it made at its first incidence; so that the lines qt, rv, sw, in which they come from the drop, will be parallel to each other, as well as the lines ab, cd, ef, in which they came to the drop. By this means these rays to a spectator commodiously situated will become visible. But all the other rays, as well those nearer the center of the drop xy, za, as those more remote from it ߠ?, de, will be reflected in lines not parallel to the lines hm, kn, lo; namely, the ray xy, in the line ??, the ray ?a in the line ??, the ray ߠ? in the line ?, and the ray de in the line ??. Whence these rays after their next reflection and subsequent refraction will be scattered from the forementioned rays, and from one another, and by that means become invisible.

4. It is farther to be remarked, that if in the first case the incident rays ab, cd, ef, and their correspondent emergent rays hm, kn, lo, are produced till they meet, they will make with each other a greater angle, than any other incident ray will make with its corresponding emergent ray. And in the latter case, on the contrary, the emergent rays qt, rv, sw make with the incident rays an acuter angle, than is made by any other of the emergent rays.

5. Our author delivers a method of finding each of these extream angles from the degree of refraction being given; by which method it appears, that the first of these angles is the less, and the latter the greater, by how much the refractive power of the drop, or the refrangibility of the rays is greater. And this last consideration fully compleats the doctrine of the rainbow, and shews, why the colours of each bow are ranged in the order wherein they are seen.

6. Suppose A (in fig. 162.) to be the eye, B, C, D, E, F, drops of rain, Mn, Op, Qr, St, Vw parcels of rays of the sun, which entring the drops B, C, D, E, F after one reflection pass out to the eye in A. Now let Mn be produced to ? till it meets with the emergent ray likewise produced, let Op produced meet its emergent ray produced in ?, let Qr meet its emergent ray in ?, let St meet its emergent ray in , and let Vw meet its emergent ray produced in ?. If the angle under M?A be that, which is derived from the refraction of the violet-making rays by the method we have here spoken of, it follows that the violet light will only enter the eye from the drop B, all the other coloured rays passing below it, that is, all those rays which are not scattered, but go out parallel so as to cause a sensation. For the angle, which these parallel emergent rays makes with the incident in the most refrangible or violet-making rays, being less than this angle in any other sort of rays, none of the rays which emerge parallel, except the violet-making, will enter the eye under the angle M?A, but the rest making with the incident ray M? a greater angle than this will pass below the eye. In like manner if the angle under O?A agrees to the blue-making rays, the blue rays only shall enter the eye from the drop C, and all the other coloured rays will pass by the eye, the violet-coloured rays passing above, the other colours below. Farther, the angle Q?A corresponding to the green-making rays, those only shall enter the eye from the drop D, the violet and blue-making rays passing above, and the other colours, that is the yellow and red, below. And if the angle SA answers to the refraction of the yellow-making rays, they only shall come to the eye from the drop E. And in the last place, if the angle V?A belongs to the red-making and least refrangible rays, they only shall enter the eye from the drop F, all the other coloured rays passing above.

7. But now it is evident, that all the drops of water found in any of the lines A?, A?, A, A?, whether farther from the eye, or nearer than the drops B, C, D, E, F, will give the same colours as these do, all the drops upon each line giving the same colour; so that the light reflected from a number of these drops will become copious enough to be visible; whereas the reflection from one minute drop alone could not be perceived. But besides, it is farther manifest, that if the line A? be drawn from the sun through the eye, that is, parallel to the lines Mn, Op, Qr, St, Vw, and if drops of water are placed all round this line, the same colour will be exhibited by all the drops at the same distance from this line. Hence it follows, that when the sun is moderately elevated above the horizon, if it rains opposite to it, and the sun shines upon the drops as they fall, a spectator with his back turned to the sun must observe a coloured circular arch reaching to the horizon, being red without, next to that yellow, then green, blue, and on the inner edge violet; only this last colour appears faint by being diluted with the white light of the clouds, and from another cause to be mentioned hereafter[332].

8. Thus is caused the interior or primary bow. The drops of rain at some distance without this bow will cause the exterior or secondary bow by two reflections of the sun’s light. Let these drops be G, H, I, K, L; Xy, Za, G, ??, T? denoting parcels of rays which enter each drop. Now it has been remarked, that these rays make with the visible refracted rays the greatest angle in those rays, which are most refrangible. Suppose therefore the visible refracted rays, which pass out from each drop after two reflections, and enter the eye in A, to intersect the incident rays in p, ?, s, t, f respectively. It is manifest, that the angle under TfA is the greatest of all, next to that the angle under ?tA, the next in bigness will be the angle under GsA, the next to this the angle under Z?A, and the least of all the angle under XpA. From the drop L therefore will come to the eye the violet-making, or most refrangible rays, from K the blue, from I the green, from H the yellow, and from G the red-making rays; and the like will happen to all the drops in the lines Ap, A?, At, Af, and also to all the drops at the same distances from the line A? all round that line. Whence appears the reason of the secondary bow, which is seen without the other, having its colours in a contrary order, violet without and red within; though the colours are fainter than in the other bow, as being made by two reflections, and two refractions; whereas the other bow is made by two refractions, and one reflection only.9. There is a farther appearance in the rainbow particularly described about five years ago[333], which is, that under the upper part or the inner bow there appears often two or three orders of very faint colours, making alternate arches of green, and a reddish purple. At the time this appearance was taken notice of, I gave my thoughts concerning the cause of it[334], which I shall here repeat. Sir Isaac Newton has observed, that in glass, which is polished and quick-silvered, there is an irregular refraction made, whereby some small quantity of light is scattered from the principal reflected beam[335]. If we allow the same thing to happen in the reflection whereby the rainbow is caused, it seems sufficient to produce the appearance now mentioned.

10. Let AB (in fig. 162.) represent a globule of water, B the point from whence the rays of any determinate species being reflected to C, and afterwards emerging in the line CD, would proceed to the eye, and cause the appearance of that colour in the rainbow, which appertains to this species. Here suppose, that besides what is reflected regularly, some small part of the light is irregularly scattered every way; so that from the point B, besides the rays that are regularly reflected from B to C, some scattered rays will return in other lines, as in BE, BF, BG, BH, on each side the line BC. Now it has been observed above[336], that the rays of light in their passage from one superficies of a refracting body to the other undergo alternate fits of easy transmission and reflection, succeeding each other at equal intervals; insomuch that if they reach the farther superficies in one sort of those fits, they shall be transmitted; if in the other kind of them, they shall rather be reflected back. Whence the rays that proceed from B to C, and emerge in the line CD, being in a fit of easy transmission, the scattered rays, that fall at a small distance without these on either side (suppose the rays that pass in the lines BE, BG) shall fall on the surface in a fit of easy reflection, and shall not emerge; but the scattered rays, that pass at some distance without these last, shall arrive at the surface of the globule in a fit of easy transmission, and break through that surface. Suppose these rays to pass in the lines BF, BH; the former of which rays shall have had one fit more of easy transmission, and the latter one fit less, than the rays that pass from B to C. Now both these rays, when they go out of the globule, will proceed by the refraction of the water In the lines FI, HK, that will be inclined almost equally to the rays incident on the globule, which come from the sun; but the angles of their inclination will be less than the angle, in which the rays emerging in the line CD are inclined to those incident rays. And after the same manner rays scattered from the point B at a certain distance without these will emerge out of the globule, while the intermediate rays are intercepted; and these emergent rays will be inclined to the rays incident on the globule in angles still less than the angles, in which the rays FI and HK are inclined to them; and without these rays will emerge other rays, that shall be inclined to the incident rays in angles yet less.