A drawing tool to find reflection points
In 2013 I discovered a special drawing instrument in the Istanbul Museum of History of Science and Technology.
Thanks to the support of Prof. Dr. Fuat Sezgin of the Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, Frankfurt am Main,
I was able to investigate its origin and operation.
A publication in English and Turkish appeared in June 2019 at the The First International Prof. Dr. Fuat Sezgin History of Science in Islam Symposium.
In the Istanbul Museum of the History of Science and Technology in Islam a drawing tool has been exhibited to solve this problem in a practical way. According to the catalog, Marcolongo (1862−1943) was inspired by a drawing by Leonardo da Vinci (1452−1519). The text and the accompanying description of da Vinci is in the Codex Atlanticus. Careful study shows that the Marcolongo drawing tool is similar to that of Leonardo da Vinci, but there are also substantial differences.
The drawing instrument developed by Marcolongo is comprehensible to a wide audience. It is suitable for a practical lesson in which students produce the instrument themselves from cardboard and search for reflection points. An animation in Geogebra is on this web page.
The drawing instrument developed by Leonardo da Vinci is also suitable for a practical lesson. An animation in Geogebra is also on this web page.
Experiments of Ibn al-Haytham
Ibn al-Haytham was born in 965 in Basra in modern Iraq and died in Cairo around 1041. He wrote about many topics in astronomy, optics, and mathematics. Ibn al-Haytham has been praised for his breakthroughs in optics, as he was the first scientist who made big steps in this field since Ptolemy. Centuries later, scholars like Kepler, Snell, Beeckman and Harriot, who also worked in Optics, appreciated the mathematical character of the treatises of Ibn al-Haytham and considered him an important predecessor. According to [Smith, 2006, p. xvii] "Alhacen's experiment (to prove the equal-angles principle) is .. light-years beyond Ptolemy's in its instrumental and conceptual sophistication", and he made the major step "to determine precisely where on the surface of a convex or concave spherical mirror the radiation from a given object-point will reflect to a given center of sight."
Ptolemy had discussed the problem but he had limited himself to the easy case where the eye and the source of light are at the same distance of the center of a cylindrical mirror. The general case where the light and eye are not at the same distance of the center is far more difficult. Ibn al-Haytham solved it by a mix of practical experiments, conic sections, and rigorous mathematical proofs.
The phenomenon of light and the way in which we humans observe objects with our eyes is not straightforward. Nowadays there are plenty of textbooks and websites with educational movies, but these resources were not available in the time of Ibn al-Haytham, who had to figure out many things by himself. Moreover, he had to go against established beliefs. The working of the eye was still unknown, perfectly smooth mirrors were a luxury and cylindrical or spherical mirrors were not easy to obtain.
Ibn al-Haytham proposed the idea that light rays travel from an object to our eye in straight lines. He also understood the law of reflection stating that the angle of incidence equals the angle of reflection. He did, however, not yet have complete understanding of the laws of refraction, which specify the change of direction when for example light travels from water to air or from air to water. These rules had, however, been understood some decades earlier by al-'Ala' ibn Sahl, and were independently rediscovered in Europe by Snell and Descartes.
Nazif investigated manuscripts of Ibn al-Haythams Optics in 1942. He wrote an impressive transcription and commentary in Arabic. He added figures that help us to understand what kind of experiments Ibn al-Haythams did. Professor Sezgin reprinted Nazif's books, ordered to build a working setup based on these drawings. He also demanded a video of this setup. Everybody can watch this video at Youtube. Thanks to this video, it is immediately clear that Islamic scholars like Ibn al-Haytham should be praised for their contribution to the sciences.
Use the slider to examine the different mirrors: flat, convex circle, concave circle, convex ellipse or concave ellipse.
You can choose where the eye and the light source are situated with respect to the mirror.
You can set the viewing direction with the movable point on the circle around the eye.
With the buttons you can choose to make the drawing instrument visible or show the geometric location of the reflection points.
Text Leonardo da Vinci
Proof Leonardo da Vinci
Leonardo da Vinci provides a short mathematical proof. He claims that corresponding angles are equal to each other.
When point O has been found, to be precise, on the circumference of the mirror and point S lies on the circle around point C, the center of the mirror circle, through point B, the eye, then quadrilateral CBDS is a kite because the length of line segment BD equals that of line segment DS and the length of line segment CB equals that of line segment CS, from the nature of the construction. For every kite, diagonal OD is an angle bisector. Therefore angles BOD and SOD have the same size. Line COD passes through the center of the circle and thus is perpendicular to the tangent to that circle. Point A is on side OS. Therefore, the angles produced by lines OA and OB with the tangent line are also the same. Conclusion is therefore that in case the construction of point O has been completed, then the angle of incidence is equal to the angle of reflection. The eye in point B then sees in point O the image of the light source in point A.