Theodoric of Freiberg and Kamal al-Din al-Farisi Independently Formulate the Correct Qualitative Description of the Rainbow
The fourteenth century witnessed many important contributions to physics, including the mean speed theorem, the graphical representation of functions, and a reformulation of impetus theory that prepared the way for the concept of inertia. As significant as these were, they were primarily the result of metaphysical speculations by scholastic philosophers. As such, they did little to advance experimental methodology. However, the fourteenth century also produced what many consider the greatest successes of experimental science during the Middle Ages—a correct qualitative description of the rainbow. Surprisingly, this was discovered almost simultaneously by Theodoric of Freiberg (c. 1250-c. 1310) in Europe and Kamal al-Din al-Farisi (c. 1260-c. 1320) in Persia.
The only significant extant ancient theory of the rainbow available during the Middle Ages was that propounded by Aristotle (384-322 B.C.). According to Aristotle, the rainbow resulted from sunlight reflected from the surface of a cloud to the eye of an observer. Unlike reflection from smooth mirrors that produce images, he argued that the uneven surface of clouds could only reflect colors. Furthermore, the specific colors of the rainbow were produced by a mixture of light and darkness while their order in the bow depended on the ratio of the Sun-to-cloud to cloud-to-eye distances. Finally, the circularity of the rainbow was seen as part of the circumference of the base of a cone whose apex was the Sun and whose axis passed through the observer's eye and terminated at the center of the base.
Aristotle's account remained the most sophisticated mathematical treatment of the rainbow for almost eighteen centuries. However, his emphasis on reflection from the cloud as a whole proved a major stumbling block for later research. Ibn Sina (Avicenna, 980-1037) was one of the first to challenge this idea. He argued the cloud was not the locus of the rainbow; rather, it was the particles of moisture in front of the cloud that reflected light. Though his analysis suggested the possibility of a geometric analysis of reflection by a single raindrop, Ibn Sina failed to pursue this possibility.
Another Arabic scientist important to the story of the rainbow was Ibn al-Haytham (Alhazen, 965-c. 1040). In Kitab al-Manazir (Treasury of optics), he articulated a comprehensive scientific methodology of the logical connections between direct observations, hypotheses, and verification. This allowed the geometric analysis of physical phenomena to be translated into concrete experiments involving the manipulation of artificially created devices. Al-Haytham exploited this methodology to full effect in his optical investigations, which surpassed all previous research in the field. He conducted extensive experiments on refraction using a water-filled, spherical globe. Unfortunately, he failed to see the analogy between the glass globe and a raindrop.
Robert Grosseteste (c. 1175-1253) rejected the idea that the rainbow was formed by reflected light. He maintained that the rainbow was formed by the refraction of light rays through a cloud. The cloud acted like a lens to focus the rays on another cloud where they appeared as an image. He attributed the bow's colors to refractions through the successively denser layers of a convex cone of moisture. Though incorrect, Grosseteste's introduction of refraction into the theory of rainbows was a major advance.
Albertus Magnus (c. 1200-1280) was the first to suggest that refraction by individual drops played a role in the formation of the rainbow. He also noted that a transparent, hemispherical vessel filled with black ink projected a brightly colored semicircular arc when placed in sunlight. Albertus equated the degrees of opacity in this vessel with the different densities within Grosseteste's cone of moisture. In essence, he saw the vessel as a diminutive cloud instead of an individual raindrop.
The first quantitative contribution to rainbow studies was made by Robert Bacon (c. 1214-c. 1294). Using an astrolabe, Bacon showed that the maximum altitude of the bow is approximately 42° (the modern value is 44°). Notwithstanding this result, Bacon made no further efforts to geometrically analyze the rainbow. It seems he confused, as many before him had, Page 273 | Top of Article the physics and physiology of colors. He believed that colors produced from crystals by refraction were objective since their location did not vary when observed from different locations, as was the case with the rainbow. Since the rainbow has no definite location, Bacon felt that it could not be the result of refraction and therefore must be a subjective phenomenon.
Al-Haytham's al-Manazir exerted a strong influence over optical studies in both the Arab world and Latin-speaking West. Nevertheless, those who correctly emphasized the role of the individual raindrop in the formation of the rainbow saw no way of implementing al-Haytham's methodology. They were unable to devise a satisfactory experimental design that would allow them to analyze the optical geometry of rainbow formation, that is, until Theodoric of Freiberg (Dietrich von Freiberg) provided the key insight.
In De Iride (On the rainbow, 1304-11) Theodoric argued that a globe of water could be treated as a magnified raindrop instead of a miniature cloud. He realized, of course, that a glass sphere only approximated a raindrop. Therefore, if he was going to be able to rely on his experiments, he needed to show that effects produced by any differences could be ignored. Most significantly, a raindrop does not have a glass envelope. Consequently, light passing through the glass sphere will be refracted four times instead of twice, as occurs in a raindrop. However, since water and air refract light by about the same amount, deviations will be small and thus can be safely ignored. Similarly, Theodoric was experimenting with a stationary globe whereas actual raindrops would presumably be falling. A suggestion of Albertus dealt with this potential problem. He argued that the drops would be falling so fast and replacing each other so quickly that it would be reasonable to replace them with a stationary curtain of transparent drops. Theodoric could now feel confident that his experimental results could be applied to the rainbow.
By placing a transparent sphere of water in a darkened room and directing a beam of sunlight on to it, Theodoric was able to carefully study the paths of light rays through a raindrop. His observations indicated that the primary rainbow was formed from light rays that had been refracted twice and reflected once. An initial refraction took place when the light ray entered the raindrop. There was then a reflection inside the drop followed by another refraction as it exited. This provided an understanding of the circular form of the rainbow. It also allowed Theodoric to explain the displacement of the rainbow when viewed from a different position—as one moves, a different set of raindrops is required to form the rainbow.
Theodoric also explained the production of the secondary rainbow. The light rays forming this bow undergo two refractions and two internal reflections. The geometry of the situation immediately explained the inversion and paleness of the colors in the secondary bow. The additional reflection reverses the order of the colors as well as weakening the light.
The aspect of Theodoric's research that most clearly foreshadowed modern scientific methodology was his attempt to explain how the colors of the rainbow were generated. He employed two pairs of contraries—obscure/clear and bounded/unbounded—to explain the origin of colors. Using these principles, he formulated various hypotheses that he tested through a series of experiments. Though his proffered explanation of colors was ultimately unsuccessful, his procedures exhibited some of the interplay between theory and experiment that has become the hallmark of modern science.
The Persian scientist al-Farisi produced a correct explanation of the rainbow independent of, and possibly prior to, Theodoric. He was encouraged by his teacher, Qutb al-Din al-Shirazi (1236-1311), to make a study of al-Haytham's optical works. Al-Farisi corrected al-Haytham on some points, rejected his theories in other places, and developed his ideas when possible. In particular, he surpassed al-Haytham's work on the rainbow by modifying his methodology.
Al-Haytham's methodology required that experiments be performed directly on the objects or phenomena of interest. This was impossible for the rainbow. Still, al-Farisi thought it might be possible, under the right conditions, to construct a suitable analog, subject it to direct observation, and apply the results to the phenomenon of interest—in this case the rainbow. This then is the great achievement of al-Farisi and Theodoric, that they extended the efficacy of experimentation beyond the direct manipulation of objects of interest.
Having accepted Ibn Sina's view that the locus of the rainbow is a myriad of water droplets, al-Farisi realized that a glass sphere Page 274 | Top of Article filled with water could be used to study a raindrop. In accordance with his new methodology, he crafted experiments with this analog that led him to the correct qualitative description of the primary and secondary rainbow. Some scholars have claimed that al-Shirazi initially discovered the correct qualitative description of the rainbow while al-Farisi merely elaborated his teacher's ideas. However, Roshdi Rashed has argued convincingly against this view.
The results of al-Farisi's work remained obscure and exerted little influence on future rainbow research. Theodoric's work initially fared little better. Though his ideas were not actually lost, they had practically no impact on later-fourteenth- and even fifteenth-century optical theories about rainbows. In 1514 Jodocus Trutfetter published an account of Theodoric's theory, replete with diagrams. René Descartes (1596-1650) may have been familiar with this or some such similar treatment since many aspects of his own account of the rainbow closely resemble those of Theodoric. Regardless, Descartes applied the newly discovered law of refraction to transform Theodoric's qualitative theory into a comprehensive quantitative treatment. He was thus able to deduce the radii of both the primary and secondary bows as well as the ordering of their colors.
STEPHEN D. NORTON
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