Answer
The index of refraction of quartz for violet light is $~~1.48$
Work Step by Step
We can use the red light to find the incident angle:
$n_1~sin~\theta_1 = n_r~sin~\theta_r$
$sin~\theta_1 = \frac{n_r~sin~\theta_r}{n_1}$
$\theta_1 = sin^{-1}~(\frac{n_r~sin~\theta_r}{n_1})$
$\theta_1 = sin^{-1}~(\frac{1.45~sin~26.3^{\circ}}{1.00})$
$\theta_1 = sin^{-1}~(0.64245)$
$\theta_1 = 40^{\circ}$
We can find the index of refraction of quartz for violet light:
$n_v~sin~\theta_v = n_1~sin~\theta_1$
$n_v = \frac{n_1~sin~\theta_1}{sin~\theta_v}$
$n_v = \frac{1.00~sin~40^{\circ}}{sin~25.7^{\circ}}$
$n_v = 1.48$
The index of refraction of quartz for violet light is $~~1.48$