Vernier Scales – Definition of Least Count for a Direct Vernier What is the correct expression for the least count of a vernier scale in relation to the main (primary) scale divisions?

Introduction:
The least count (LC) expresses the smallest difference in measurement that a vernier instrument can resolve. Understanding LC is essential for reading angles and lengths accurately and for estimating observational precision in field notes.

Given Data / Assumptions:

• A direct vernier is used in which n vernier divisions coincide with (n − 1) main scale divisions.
• Let 1 MSD be the value of one main scale division.
• Let 1 VSD be the value of one vernier scale division.

Concept / Approach:

For a direct vernier, n VSD = (n − 1) MSD ⇒ 1 VSD = ((n − 1)/n) MSD. The least count equals the difference between 1 MSD and 1 VSD, which simplifies to LC = MSD / n. Therefore, one practical way to compute LC is to divide the value of one main division by the number of vernier divisions, matching the given option.

Step-by-Step Solution:

Verification / Alternative check:

Worked examples for typical theodolites: if 60 vernier divisions span 59 main divisions and 1 MSD = 20′, LC = 20′ / 60 = 20″, consistent with field instruments.

Why Other Options Are Wrong:

Sum or product have no meaning in this context; option (c) inverts the relation; 'none' is invalid since a standard expression exists.

Common Pitfalls:

Confusing LC with the smallest graduation seen on the main scale; overlooking that LC depends on both scales’ geometry.

Final Answer:

Value of one division of the main (primary) scale divided by the total number of divisions on the vernier