**Diagonal scale** is an engineering measuring instrument which is composed of a set of parallel straight lines which are obliquely crossed by another set of straight lines. Diagonal scales are used to measure small fractions of the unit of measurement.^{}

Diagonal scale follows the principle of similar triangles where a short length is divided into number of parts in which sides are proportional.^{} Divided into required number of equal parts

Diagonal scale is used in engineering to read lengths with higher accuracy as it represents a unit into three different multiple in meters, centimeters and millimeters.^{}Diagonal scale is an important part in Engineering drawings

### Vernier

A **vernier** scale is a visual aid that allows the user to measure more precisely than could be done unaided when reading a uniformly divided straight or circular measurement scale. It is a scale that indicates where the measurement lies in between two of the graduations on the main scale.

The principle of vernier is based on the fact that the eye can perceive without strain and with considerable precision when two graduations coincide to form one continuous straight line.

#### Single vernier:-

if the graduations of the main scale are numbered in one direction only, the vernier used is called a single vernier.

#### Double vernier:-

if the graduation of the main scale are numbered in both the directions, the vernier used is called double vernier.

#### least count:-

it is equal to the difference between the smallest division on the main scale as well as vernier scale.

**Types of vernier**

- Direct vernier
- Retrograde vernier

#### Direct vernier

it is the one which extends or increase in the same direction as that of the main scale and in which the smallest division on the vernier is shorter than the smallest division on the main scale.

Let the measure of the smallest main scale reading, that is the distance between two consecutive graduations (also called its *pitch*) be **S** and the distance between two consecutive Vernier scale graduations be **V** such that the length of **(n-1)** main scale divisions is equal to **n** Vernier scale divisions.Then,

*the length of (n-1) main scale divisions = the length of n vernier scale division*

or, *(n-1)S = nV*

or, *nS-S = nV*

or, *S = nS – nV*

or, *S/n = (S-V)*

or *(Pitch)/(Number of Vernier scale divisions) = (Length of one main scale division – Length of one Vernier scale division)*

*S/n* and *(S-V)* are both equal to the least count of vernier scale, and are also called the *vernier constant*.

**Retrograde vernier**

it is the one which extends or increase in opposite direction as that of the main scale and in which the smallest division of the vernier is longer than that of the smallest division on the main scale. it is constructed (n+1) divisions of the main scale are equal in length of n divisions of the vernier.

nv = (n+1)s

v = (n+1)s/n

least count = v-s =s/n

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