The formula for the refractive index of a prism is: $$\mu = \frac{\sin \left(\frac{A + D_m}{2}\right)}{\sin (A/2)}$$
However, this requires me to find out the angle of minimum deviation ($D_m$). This would require me to take several readings, plot a graph, and then find the minimum point of the graph.
If I have two readings of the angle of emergence and angle of incidence, is there a way I can find the refractive index of the prism?
The following diagram should help:
We know of course the relationship between $\alpha$ and $\beta$ because the angle $A$ is known: $$\alpha + \beta = A$$
Next, we have the relationship between $i$ and $\alpha$ from Snell's law:
$$\frac{\sin{i}}{n}=\sin\alpha$$ and similarly for $\beta$:
$$\frac{\sin{e}}{n}=\sin\beta$$
This gives you three equations with three unknowns: $\alpha,~\beta,~n$.
I would recommend solving this numerically, or graphically. Simple example of a graphical solution:
I calculated the expected exit angle given an input angle of 45° and a refractive index of 1.543 - then calculated what exit angle I would get if I didn't know the refractive index. The error goes to zero when you have the right index...
Python code used to generate this:
# prism simulation
from math import pi, asin, sin
import numpy as np
import matplotlib.pyplot as plt
A = 60*pi/180. # 60 degrees, in radians
# pick an angle of incidence
i = pi/4
# pick a refractive index
n = 1.543
# compute the exit angle
alpha = asin(sin(i)/n)
beta = A - alpha
e = asin(sin(beta)*n)
print 'exit angle is %.1f deg'%(e*180/pi)
# calculate n graphically from just i and e
na = np.arange(1.4,1.6,0.001) # possible values
# what would the intermediate values alpha, beta be?
aa = np.arcsin(np.sin(i)/na)
bb = A - aa
ee = np.arcsin(np.sin(bb)*na)
plt.figure()
plt.plot(na, 180*(ee-e)/pi)
plt.plot([n,n],[-20,20])
plt.plot([1.4,1.6],[0,0])
plt.xlabel('possible value of n')
plt.ylabel('error in exit angle')
plt.title('graphically estimating n')
plt.show()
Note that my answer exploits a loophole in your question (namely that you haven't said that the 2 known values of the required angles are random):
If at a particular angle of incidence $i$, the angle of emergence $e$ is equal to $i$, then we have the condition for minimum deviation. Then,
$$ \delta_{min}=2i-A$$, where $A$ is the angle of the prism. Then you can use the relation you mentioned to calculate the refractive index $\mu$ of the prism.
On a side note, and as an answer which does not exploit a loophole in the question, you would be better off using a standard source of light such as a sodium-D lamp (589 nm) and tilting the source until you get a condition for grazing emergence. Then, you can use geometry to find the angle of incidence inside the prism, which will be equal to its critical angle $i_c$, and, as we know,
$$\sin i_c= \frac {1}{\mu}$$
