# Prologue

Two dramatically different philosophical approaches to science were developed in the field of classical mechanics during the 17\(^{th}\) - 18\(^{th}\) centuries. This time period coincided with the Age of Enlightenment in Europeduring which remarkable intellectual and philosophical developments occurred. This was a time when both

philosophical and causal arguments were equally acceptable in science, in contrast with current convention where there appears to be tacit agreement to discourage use of philosophical arguments in science.

%% underconstruction %% Figure 1: Vectorial and variational representation of Snell’s Law for refraction of light.

\(\color{MidnightBlue}\textbf{Snell’s Law}\): The genesis of two contrasting philosophical approaches to science relates back to early studies of the reflection and refraction of light. The velocity of light in a medium of refractive index \(\textit{n}\) equals \(\mathit{v = \frac{c}{n} }\). Thus a light beam incident at an angle \( \theta_1 \) to the normal of a plane interface between medium 1 and medium 2 is refracted at an angle \( \theta_2 \) in medium 2 where the angles are related by Snell’s Law.

\begin{equation}\label{eq:Snell's Law}\tag{Snell's Law}

\frac{sin \theta_1}{sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}

\end{equation}

Ibn Sahl of Bagdad (984) first described the refraction of light, while Snell (1621) derived his law mathematically. Both of these scientists used the "vectorial approach" where the light velocity \( \mathit{v} \) is considered to be a vector pointing in the direction of propagation.

\(\color{MidnightBlue}\mathbf{Fermat’s \ Principle:}\) Fermat’s principle of least time (1657), which is based on the work of Hero of Alexandria (∼ 60) and Ibn al-Haytham (1021), states that "light travels between two given points along the path of shortest time," where the transit time \( \tau \)of a light beam between two locations \(\mathit{v}\) and \(\mathit{B}\) in a medium with position-dependent refractive index \(\mathit{n(s)}\) given by

\begin{equation}

\tag{Fermat's Principle}

\mathrm{\tau=\int_{t_A}^{t_B}dt=\frac{1}{c}\int_A^Bn(s)}

\label{eq:Fermat's Principle}

\end{equation}