To determine refractive index of glass slab using a travelling microscope, minimum number of readings required are:

The refractive index of a transparent slab is usually obtained with a travelling microscope by comparing the real thickness of the slab with its apparent thickness when viewed through the same slab.

First, we write the basic relation that connects the required quantities:

$$\mu = \dfrac{t}{d},$$

where $$\mu$$ is the refractive index of the material, $$t$$ is the actual (real) thickness of the glass slab, and $$d$$ is its apparent thickness as seen through the slab.

To evaluate $$t$$, the microscope is focused successively on the upper surface and the lower surface of the glass. If the microscope reading when it is in focus on the upper surface is $$R_1$$ and the reading when it is in focus on the lower surface is $$R_2$$, then, because the vertical scale on the microscope gives the distance travelled by its objective, we have

$$t = |R_2 - R_1|.$$

Next, to evaluate the apparent thickness $$d$$, the slab is kept on a paper having a fine cross-mark. The microscope is again focused on the upper surface and the corresponding reading is already known to be $$R_1$$. Without disturbing the slab, the microscope is then lowered until the image of the cross-mark, seen through the glass, comes into sharp focus. Let the reading in this second position be $$R_3$$. The apparent depth of the cross below the upper surface is therefore

$$d = |R_3 - R_1|.$$

Thus, to arrive at $$\mu$$, the experimenter needs the three independent scale readings $$R_1, R_2,$$ and $$R_3$$. Substituting these into the formula gives

$$\mu = \dfrac{|R_2 - R_1|}{|R_3 - R_1|}.$$

We note carefully that the reading $$R_1$$ is common to both the real and the apparent thickness determinations, so it is not necessary to take any additional reading beyond these three. Hence, the minimum number of microscope readings that must be recorded is three.

Hence, the correct answer is Option C.