Developing Ideas of Refraction, Lenses and Rainbow Through the Use of Historical Resources

Abstract

The paper examines different ways of using historical resources in teaching refraction related subjects. Experimental procedures can be taught by using Ptolemy’s and Al Haytham’s methods. The student can check the validity of the approximations or rules which were presented by different people. The interpretation of the relations is another subject. Refraction phenomena were interpreted either by the principle of least time or by particles or by waves. The law of refraction can be used as an example of a law which was discovered but put aside. The use of the law to construct lenses can be seen in Ibn Sahl’s hyperbolical lenses. Al Farisi’s method of “cones” is used for the interpretation of the rainbow. Al Farisi’s model was discovered again by Descartes. These models were not able to explain the supernumerary arcs. For this reason a simple wave model is presented. The models proposed by Al Haytham of atmospheric refraction can be used to show that refraction actually cannot be considered as the cause of the change of the size of the moon. Finally Huygens model of refraction in the atmosphere is used to introduce the wave fronts as more fundamental than rays.

Appendix A: Newton’s Analogy

The incident ray is GH, the refracted ray is IK (Figure 8) Then HM = $v_{i}\cdot\hbox{t}$ and MI = $1/2\cdot\hbox{a}\cdot\hbox{t}^{2}$. From this we get:

$$ \hbox{HM}^{2}=\hbox{MI}\cdot 2\cdot\hbox{v}_{i}{}^{2}/\hbox{a}$$

In the original derivation in Principia, Newton used the fact that HI is a parabola. It can be proven by elementary Analytical Geometry that point L has coordinates$x_{\rm L}=(x_{\rm H}+x_{\rm M})/2$,$y_{\rm L}=a x_{\rm H}\cdot x_{\rm M}$ and from this he draws a circle (L,LI). O is the middle of MR. Then since the “constant of inversion” of point M concerning the circle is equal to MN·MI = MQ·MP = ML $^{2}-LI^{2}$. Since ML=HM/2 we get

$$\eqalign{ LI^{2}/ML^{2}=4 LI^{2}/HM^{2}=1-4MN \cdot MI/HM^{2}=1-2\cdot a\cdot \cr MN/v_{i}{}^{2}=(v_{i}{}^{2}-2\cdot a\cdot MN)/v_{i}{}^{2}=v_{r}{}^{2}/v_{i}{}^{2}} $$

From elementary geometry: LI/ML = $\sin(\theta_{i})/\sin(\theta_{r}) = v_{r}/v_{i}$.

This is Descartes result.