Rainbow, an arch of concentric colored bands, visible usually on a portion of sky overspread with falling rain drops, and always on that side of the observer opposite to the place from which the sun or moon is shining at the time. When the field of falling drops is large, and the illumination thrown on it is bright, a second bow, exterior to and concentric with the first, appears. The inner, or most usual, is termed the primary, the outer the secondary bow. Each shows the same colors, and in the same succession, as those obtained in decomposing a beam of sunlight by means of a dispersing prism of glass; but in the two bows the colors lie in opposite order; in the primary the red is outermost, in the secondary innermost. The primary is always the brighter, and decidedly the narrower. When the light is abundant, this bow is often accompanied by successive bands of red and green, lying just within it or overlapping its violet edge, concentric with it, but extending through parts of its course only, and especially where it nears the horizon; these are called supernumerary bows.

The common centre of the two bows is always in the direction of the antisolar point; so that, of course, the rainbow rises at the same rate as the sun declines, or declines if the sun is rising. - The conditions requisite to produce the rainbow have been in a general way understood from an early period, though its causes were not. The earliest known attempt at an explanation of it is that of Aristotle. He observed that from a glass globe filled with water, and set in the sun, certain colors were always returned at certain angles with the course of the sun's beams; and he properly explained the circular form of the bow, by saying that if the sunbeam passing through the observer's eye be taken as an axis, and the globe be revolved round this axis, and at the same distance from it in all parts of its course, the same colors, preserving their angle with the direction of the sunbeams or of the axis, would be visible through all parts of this course; and hence it followed that a rainbow would result if there were globes enough, and so placed as to reflect colors at the same time from all parts of an arc of such a circle.

The colors were supposed to be merely reflected from the globe, or (in the sky) from the drop of water until Fleischer of Breslau (1571), concluding that reflected light does not give colors, stated as a consequence that the rays must enter the drops. Of the light falling on the presented side of the drops, of course part will be reflected, but another part will enter and be refracted at the same time; striking on the inner opposite surface of the drop, part of this beam will emerge and escape, while another part will be reflected; and on again striking the side of the drop toward the spectator, though a portion of this residue of the first beam undergoes a second reflection, another portion emerges, again refracted, and, if at a proper angle, then passes to the eye. Kepler agreed in this view, but erred in supposing the entering light to be that of rays grazing or tangent to the upper sides of the drops. Antonio de Dominis, in 1611, carefully repeated the experiments with the glass sphere filled with water, showing in sunlight very vivid colors to a great distance, and each at an angle of its own.

Descartes showed: 1, why there must be on the illuminated field of falling drops a circular belt of colors bright enough to be seen, and always of a definite diameter; and 2, that the colors are in separate bands or stripes in this, because they are not equally refracted. He gave the reasons why the colors must be just where they were, and in bands just so broad, if they all appeared; he could not tell why they must all appear. This element Newton supplied, when he discovered (1666) that sunlight is decomposable into a fixed number of different colored rays, refracted or bent at the same time in different but definite degrees, so that they must appear, under given circumstances, separated just so much, and always in the same successive order. This result will follow, then, whether sunlight is dispersed by prisms or by transparent spheres, as water drops. The mathematical theory, which belongs to Descartes, may be found in the higher text books of optics, and is illustrated by the accompanying diagram taken from Deschanel's "Natural Philosophy." If a ray of light pass through the centre of a sphere or drop, its course is in an axis of the sphere or drop; it is not refracted. A ray parallel with this, and very near it, is refracted within the drop, toward this axis, but very slightly.

Other rays, further and further from the axis, are refracted more and more toward it, but yet so as to fall, by lessening degrees, further from it on the inner or second surface of the drop; until, as Descartes proved, a ray, S b or S a, entering the upper side of the drop, when this is above the eye, and at a point for which its angle of incidence is 60°, will strike on the inner surface as far as any ray can do from the axis; the rays incident at greater angles than this, up to 90°, deviating again toward the axis. Of course, near this limit, the deviation is very slight for rays coming on either side, so that much more light within the drop will be accumulated just at this point of the second surface than at any. other; and though part of it emerges here, a sufficient quantity is reflected, and that in rays which preserve a parallel course (b O or a O), after leaving the drop in the direction toward the spectator, to form a compact, parallel beam, bright enough to affect the eye at a great distance.

The apparent radii of the arcs constituting the rainbow are constant, or nearly so; they are expressed by the angles between the axis OZ and the lines Oa, Ob, etc, and are as follows: in the primary bow, for the violet 40° 17', for the red 42° 2'; in the secondary bow, for the red 50° 57', for the violet 54° 7'. A tertiary bow, formed by rays that have been thrice reflected within the raindrops, is possible at a distance of about 43° 50' from the sun; but this is very rarely visible, owing to its faintness and other causes. From the above explanation, the following consequences are obvious: that the ordinary rainbows must be on the side of the observer opposite the sun; that their centres must be directly opposite the sun; that they must move with the motion of the sun, declining in the morning, and rising if seen at evening; that when the sun and the observer are in the same horizontal plane, as at sunset, the bows will be semicircles, and their altitudes then about 42° and 54°; that they can never approach nearer than this to the zenith, unless the observer be on an elevated position, so that the sun can shine from below the horizontal plane in which he is; that at the tops of high mountains they may be seen as complete circles; and that, to one at the ordinary level, in the low and middle latitudes, they are never seen between about 9 o'clock in the morning and 3 o'clock in the afternoon; while in higher latitudes, where the sun is always very low in the sky, they may occur even at midday.

If the rain is near, the bows may sometimes be seen prolonged upon the landscape. The small water drops constituting spray may afford a rainbow; hence it is seen in the mist arising near cataracts, and, because near, is then small, and may appear as a complete circle. A partial bow may be observed at times in drops of dew or rain upon herbage or grass. The formation of the supernumerary bows was explained by Young (1804), as due to interference of sets of rays emerging at angles very nearly those of the proper colors of the bows. Biot, and afterward Brewster, have shown that in all rainbows the light is polarized in the radial planes passing through the axis O Z, and hence polarized by refraction and reflection. - The lunar rainbow is usually single, the primary bow only, and is often white; when colored, it is but faintly so. - When the drops of rain are exceedingly fine, as in the case of clouds and fog, the rainbow proper is replaced by bows formed by the reflection and interference of light from these fine particles. The laws of these fog bows are deducible from the same principles that have served to explain the rainbow.

The phenomena themselves are exceedingly brilliant; they were observed by Sykes in 1829 (see "Philosophical Transactions," 1835), but far more perfectly by the aeronauts of the past few years; beautiful examples are recorded in Glaisher's "Travels in the Air" (London, 1870). - The floating ice spiculae or crystals that compose those higher clouds called cirri affect the solar rays even more curiously than the spherical drops of water, causing the varied phenomena of parhelia, all of which are explainable on principles not materially different from those that apply to the rainbow proper.