Emus tenui aqua tegitur et fracti speciem reddit — Seneca, 63 AD

Painting by an unknown Baroque artist. Claudius Ptolemaeus was born in Egypt around AD 85 and worked in Alexandria as a scholar in the Museum. The instrument in his right hand is a cross-staff, which is used to measure angles between celestial objects / the horizon. The cross-staff was probably not invented until the 14th century.
Angle of refraction (thetaR) as a function of angle of incidence (thetaI) for air-water interface based on Ptolemy’s calculations (red line / thetaRpt) v. Snell’s Law (blue line / thetaR).

From Steven R. Wilk at http://www.osa-opn.org/Content/ViewFile.aspx?id=5392:

“I had learned, as I think every optics student has, that the law of refraction was discovered by Dutch mathematician Willebrord van Roijen Snell in 1621 and first published in Christiaan Huygens 1703 book, “Dioptrica.” The law was not published by Snell during his lifetime and was independently discovered by René Descartes; it is thus known in France as Descartes’s law. By means of this powerful law, Descartes was able to calculate the location of the rainbow and Newton was able to derive the laws of imaging with lenses. Because of the careful experimental work of Snell, Descartes and others, modern optics took root in the 17th century.

“So I was amazed to discover that mathematician, astronomer and physicist Claudius Ptolemy had already performed a series of careful experiments in the first century C.E. to determine the rules of refraction. What’s more, Ptolemy’s was a quantitative study, and a startlingly modern one at that, performed just as it is today in many undergraduate optics labs. Ptolemy constructed a straightforward goniometer, marking off the degrees along the edge and placing the center at the interface between two dielectric media. He examined refraction at the interface between air and water, between glass and air, and between glass and water. He varied the angle of incidence between 10 degrees and 80 degrees in units of 10 degrees (measured from the normal to the interface) and measured the corresponding angle of refraction.

“George Sarton, in his “Introduction to the History of Science,” called Ptolemy’s work “the most remarkable experimental research of antiquity.” Ptolemy’s work certainly does follow the model of experimental science we’ve been brought up to revere. He made an observation (the apparently “broken” oar), hypothesized a cause (change of angle at the water-air interface), arranged an experiment in exemplary fashion, made his observations and came up with a result.

“Several writers take Ptolemy to task for not having determined the correct relationship between the sines of the angles of incidence and refraction. This is somewhat ironic, since Ptolemy compiled what is essentially a widely used table of sines. What’s more, it’s by no means obvious that the sines of the angles would be the functions to have a linear relationship. Moreover, Ptolemy’s tables did not give the sines of the angles directly, but rather the lengths of the chords of those angles, measured on a circle having a radius of 60 units. The chord of an angle is proportional to the sine of half the angle, and this makes the relationship even less obvious.

“Ptolemy’s work had its greatest influence through the Arabic students of optics, who knew of it indirectly through the work of Ibn al-Haitham (better known as Alhazen). Even with this audience, however, it did not find full expression. Among the lost parts of Ptolemy’s “Optics” is his explanation of the rainbow. The Arabic opticists Qutb al-Din al-Shirazi (1236-1311) and his student Kamal al-Din al-Farisi (circa 1320) both studied the rainbow, producing surprisingly modern results. They modeled the raindrop as a sphere of water and experimented with a glass sphere filled with water, following the beam of light as it refracted, reflected once inside, then refracted upon leaving the drop. They correctly explained both the primary and secondary rainbow in this way, and made the first observation of a tertiary rainbow. At almost precisely the same time, a French-German monk named Theodoric of Freibourg was working along the same lines in Europe. His drawings have come down to us (al-Shirazi’s and al-Farisi’s, sadly, have not), and his pictures of a primary or a secondary rainbow could have been taken from a modern optics text, or from Descartes’ work on the rainbow.

“Descartes, you may recall, used Snell’s law to imagine the trajectories of rays entering a drop of water at different distances from the center and, noting that the rainbows occurred at a maxima or a minima in the angle of refraction, calculated for the first time the angle of the rainbow. If they had been armed with Ptolemy’s formula, al-Farisi, al-Shirazi or Theodoric, who knew the path of the ray through the drop, could have done the same 300 years earlier. Moreover, they would have obtained the correct result. Unlike the paraxial optics examples I noted above, which depend on the values at small angles (where Ptolemy’s insistence on constant second differences led him into error), the rainbow angle is calculated on angles where Ptolemy’s results are almost perfectly correct (despite his having fudged the data). The quadratic formula you derive from Ptolemy’s data predicts that the rainbow lies at 42 degrees from the anti-solar point. The modern geometrical optics result, using Snell’s law, predicts 42.5 degrees.

“In other words, we came extremely close to having a correct theoretical explanation for the rainbow several centuries in advance of its actual discovery, and only missed it (arguably) because Ptolemy’s book didn’t quite last long enough or wasn’t popular enough to reach the proper hands.”