# 1911 Encyclopædia Britannica/Rainbow

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​RAINBOW, formerly known as the iris, the coloured rings seen in the heavens when the light from the sun or moon shines on falling rain; on a smaller scale they may be observed when sunshine falls on the spray of a waterfall or fountain. The bows assume the form of concentric circular arcs, having their common centre on the line joining the eye of the observer to the sun. Generally only one bow is clearly seen; this is known as the primary rainbow; it has an angular radius of about 41°, and exhibits a fine display of the colours of the spectrum, being red on the outside and violet on the inside. Sometimes an outer bow, the secondary rainbow, is observed; this is much fainter than the primary bow, and it exhibits the same play of colours, with the important distinction that the order is reversed, the red being inside and the violet outside. Its angular radius is about 57°. It is also to be noticed that the space between the two bows is considerably darker than the rest of the sky. In addition to these prominent features, there are sometimes to be seen a number of coloured bands, situated at or near the summits of the bows, close to the inner edge of the primary and the outer edge of the secondary bow; these are known as the spurious, supernumerary or complementary rainbows.
The formation of the rainbow in the heavens after or during a shower must have attracted the attention of man in remote antiquity. The earliest references are to be found in the various accounts of the Deluge. In the Biblical narrative (Gen. ix. 12–17) the bow is introduced as a sign of the covenant between God and man, a figure without a parallel in the other accounts. Among the Greeks and Romans various speculations as to the cause of the bow were indulged in; Aristotle, in his Meteors, erroneously ascribes it to the reflection of the sun’s rays by the rain; Seneca adopted the same view. The introduction of the idea that the phenomenon was caused by refraction is to be assigned to Vitellio. The same conception was utilized by Theodoric of Vriberg, a Dominican, who wrote at some time between 1304 and 1311 a tract entitled De radialibus impressionibus, in which he showed how the primary bow is formed by two refractions and one internal reflection; i.e. the light enters the drop and is refracted; the refracted ray is then reflected at the opposite surface of the drop, and leaves the drop at the same side at which it enters, being again refracted. It is difficult to determine the influence which the writings of Theodoric had on his successors; his works were apparently unknown until they were discovered by G. B. Venturi at Basel, partly in the city library and partly in the library of the Dominican monastery. A full account, together with other early contributions to the science of light, is given in Venturi’s Commentari sopra la storia de la Teoria del Ottica (Bologna, 1814). John Fleischer (sometimes incorrectly named Fletcher), of Breslau, propounded the same view in a pamphlet, De iridibus doctrina Aristotelis et Vitellonis (1574); the same explanation was given by Franciscus Maurolycus in his Photismi de lumine et umbra (1575).
The most valuable of all the earlier contributions to the scientific explanation of rainbows is undoubtedly a treatise by Marco Antonio de Dominis (1566–1624), archbishop of Spalatro. This work, De radiis visūs et lucis in vitris perspectivis et iride, published at Venice in 1611 by J. Bartolus, although written some twenty years previously, contains a chapter entitled “Vera iridis tota gene ratio explicatur,” in which it is shown how the primary bow is formed by two refractions and one reflection, and the secondary bow by two refractions and two reflections. Descartes strengthened these views, both by experiments and geometrical investigations, in his Meteors (Leiden, 1637). He employed the law of refraction (discovered by W. Snellius) to calculate the radii of the bows, and his theoretical angles were in agreement with those observed. His methods, however, were not free from tentative assumptions, and were considerably improved by Edmund Halley (Phil. Trans., 1700, 714). Descartes, however, could advance no satisfactory explanation of the chromatic displays; this was effected by Sir Isaac Newton, who, having explained how white light is composed of rays possessing all degrees of refrangibility, was enabled to demonstrate that the order of the colours was in perfect accord with the requirements of theory (see Newton’sOpticks, book i. part 2, prop. 9).
The geometrical theory, which formed the basis of the investigations of Descartes and Newton, afforded no explanation of the supernurnerary bows, and about a century elapsed before an explanation was forthcoming. This was given by Thomas Young, who, in the Bakerian lecture delivered before the Royal Society on the 24th of November 1803, applied his principle ​of the interference of light to this phenomenon. His not wholly satisfactory explanation was mathematically examined in 1835 by Richard Potter (Camb. Phil. Trans., 1838, 6, 141), who, while improving the theory, left a more complete solution to be made in 1838 by Sir George Biddell Airy (Camb. Phil. Trans., 1838, 6, 379).
The geometrical theory first requires a consideration of the path of a ray of light falling upon a transparent sphere. Of the total amount of light falling on such a sphere, part is reflected or scattered at the incident surface, so rendering the drop visible, while a part will enter the drop. Confming our Geometrical theory.attention to a ray entering in a principal plane, we will determine its deviation, i.e. the angle between its directions of incidence and emergence, after one, two, three or more internal reflections.
* Fig. 1.
Let EA be a ray incident at an angle i (fig. 1); let AD be the refracted ray, and r the angle of refraction. Then the deviation experienced by the ray at A is i-r. If the ray suffers one internal reflection at D, then it is readily seen that, if DB be the path of the reflected ray, the angle ADB equals 2r, i.e. the deviation of the ray at D is π−2r. At B, where the ray leaves the drop, the deviation is the same as at A, viz. i-r. The total deviation of the ray is consequently given by D=2(i-r)+π−2r.
Similarly it may be shown that each internal reflection introduces a supplementary deviation of π−2r; hence, if the ray be reflected n times, the total deviation will be D=2(i-r)+n(π−2r).
The deviation is thus seen to vary with the angle of incidence; and by considering a set of parallel rays passing through the same principal plane of the sphere and incident at al angles, it can be readily shown that more rays will pass in the neighbourhood of the position of minimum deviation than in any other position (see Refraction). The drop will consequently be more intensely illuminated when viewed along these directions of minimum deviation, and since it is these rays with which we are primarily concerned, we shall proceed to the determination of these directions.
Since the angles of incidence and refraction are connected by the relation sin i=μ sin r (Snell’s Law), μ being the index of refraction of the medium, then the problem may be stated as follows: to determine the value of the angle i which makes D=2(i-r)+n(π−2r) a maximum or minimum, in which i and r are connected by the relation sin i=μ sin r, μ being a constant. By applying the method of the differential calculus, we obtain cos i=√(μ$^2$−1)/(n$^2$+2n) as the required value; it may be readily shown either geometrically or analytically that this isia minimum. For the angle i to be real, cos i must be a fraction, that is n$^2$+2n>μ$^2$−1, or (n+1)$^2$>μ$^2$. Since the value of μ for water is about 4/3, it follows that n must be at least unity for a rainbow to be formed; there is obviously no theoretical limit to the value of n, and hence rainbows of higher orders are possible.
So far we have only considered rays of homogeneous light, and it remains to investigate how lights of varying refrangibilities will be transmitted. It can be shown, by the methods of the differential calculus or geometrically, that the deviation increases with the refractive index, the angle of incidence remaining constant. Taking the refractive index of water for the red rays as 108/81, and for the violet rays as 109/81, we can calculate the following values for the minimum deviations corresponding to certain assigned values of n.
| n | Red. | Violet. |
| 1 2 3 4 | π –  42°.1 2π – 129°.2 3π – 231°.4 4π – 317°.07 | π –  40°.22 2π – 125°.48 3π – 227°.08 4π – 310°.07 |
To this point we have only considered rays passing through a principal section of the drop; in nature, however, the rays impinge at every point of the surface facing the sun. It may be readily deduced that the directions of minimum deviation for a pencil of parallel rays lie on the surface of cones, the semi-vertical angles of which are equal to the values given in the above table. Thus, rays suffering one internal reflection will all lie within a cone of about 42°; in this direction the illumination will be most intense; within the cone the illumination will be fainter, while, without it, no light will be transmitted to the eye.
Fig. 2.
Fig. 2 represents sections of the drop and the cones containing the minimum deviation rays after 1, 2, 3 and 4 reflections; the order of the colours is shown by the letters R (red) and V (violet). It is apparent, therefore, that all drops transmitting intense light after one internal reflection to the eye will lie on the surfaces of cones having the eye for their common vertex, the line joining the eye to the sun for their axis, and their semi-vertical angles equal to about 41° for the violet rays and 43° for the red rays. The observer will, therefore, see a coloured band, about 2° in width, and coloured violet inside and red outside. Within the band, the illumination will be faint; outside the band there will be perceptible darkening until the second bow comes into view. Similarly, drops transmitting rays after two internal reflections will be situated on covertical and coaxial cones, of which the semi-vertical angles are 51° for the red rays and 54° for the violet. Outside the cone of 54° there will be faint illumination; within it, no secondary rays will be transmitted to the eye. We thus see that the order of colours in the secondary bow is the reverse of that in the primary; the secondary is half as broad again (3°), and is much fainter, owing to the longer path of the ray in the drop, and the increased dispersion.
Similarly, the third, fourth and higher orders of bows may be investigated. The third and fourth bows are situated between the observer and the sun, and hence, to be viewed, the observer must face the sun. But the illumination of the bow is so weakened by the repeated reflections, and the light of the sun is generally so bright, that these bows are rarely, if ever, observed except in artificial rainbows. The same remarks apply to the fifth bow, which differs from the third and fourth in being situated in the same part of the sky as the primary and secondary bows, being just above the secondary.
The most conspicuous colour band of the principal bows is the red; the other colours shading off into one another, generally with considerable blurring. This is due to the superposition of a great number of spectra, for the sun has an appreciable apparent diameter, and each point on its surface gives rise to an individual spectrum. This overlap ing may become so pronounced as to produce a rainbow in which colour is practically absent; this is particularly so when a thin cloud intervenes between the sun and the rain, which has the effect of increasing the apparent diameter of the sun to as much as 2° or 3°. This phenomenon is known as the “white rainbow” or “Ulloa’s Ring or Circle,” after Antonio de Ulloa.
We have now to consider the so-called spurious bows which are sometimes seen at the inner edge of the primary and at the outer edge of the secondary bow. The geometrical theory can afford no explanation of these coloured bands, and it has been shown that the complete phenomenon of the rainbow Physical theory. is to be sought for in the conceptions of the wave theory of light. This was first suggested by Thomas Young, who showed that the rays producing the bows consisted of two systems, which, although emerging in parallel directions, traversed different paths in the drop. Destructive interference between these superposed rays will therefore occur, and, instead of a continuous maximum illumination in the direction of minimum deviation, we should expect to find alternations of brightness and darkness. The later investigations of Richard Potter and especially of Sir George Biddell Airy have proved the correctness of Young’s idea. The mathematical discussion of Airy showed that the primary rainbow is not situated directly on the line of minimum deviation, but at a slightly greater value; this means that the true angular radius of the bow is a little less than that derived from the geometrical theory. In the same way, he showed that the secondary bow has a greater radius than that previously assigned to it. The spurious bows he showed to consist of a series of dark and bright bands, whose distances from the principal bows vary with the diameters of the raindrops. The smaller the drops, the greater the distance; hence it is that the spurious bows are generally only observed near the summits of the bows, where the drops are smaller than at any lower altitude. In Airy’s investigation, and in the extensions by Boitel, J. Larmor, E. Mascart and L. Lorentz, the source of light was regarded as a point. In nature, however, this is not realized, for the sun has an appreciable diameter. Calculations taking this into account have been made by J. Pernter (Neues ilber den Regenbogen, Vienna, 1888) and by K. Aichi and T. Tanakadate (Jour. College of Science, Tokyo, 1906, vol. xxi. art. 3).
Experimental confirmation of Airy’s theoretical results was afforded in 1842 by William Hallows Miller (Camb. Phil. Trans. vii. 277). A horizontal pencil of sunlight was admitted by a vertical slit, and then allowed to fall on a column of water supplied by a jet of about Blyth of an inch in diameter. Primary, secondary and spurious bows were formed, and their radii measured; a comparison of these observations exhibited agreement with Airy’s analytical values. Pulfrich (Wied. Ann., 1888, 33, 194) obtained similar results by using cylindrical glass rods in place of the column of water.
Lunar rainbows. The moon can produce rainbows in the same manner as the sun. The colours are much fainter, and according to Aristotle, who claims to be the first observer of this phenomenon, the lunar bows are only seen when the moon is full.
Marine rainbow is the name given to the chromatic displays formed by the sun's rays falling on the spray drawn up by the wind playing on the surface of an agitated sea.
Intersecting rainbows are sometimes observed. They are formed by parallel rays of light emanating from two sources, as, for example, the sun and its image in a sheet of water, which is situated between the observer and the sun. In this case the second bow is much fainter, and has its centre as much above the horizon as that of the direct system is below it.
References.—For the history of the theory of the rainbow, see G. B. Venturi, Commentari sopra la storia de la teoria del Ottica (Bologna, 1814); F. Rosenberger, Geschichte der Physik (1882–90). The geometrical and physical theory is treated in T. Preston’s Theory of Light; E. Mascart’s Traité d'optique (1899–1903); and most completely by J. Pernter in various contributions to scientific journals and in his Meteorologische Optik (1905–9).