Plain, diagonal and vernier scales

A scale is a ruler-like graduated line used to measure ground distances on a map. Plain, diagonal and vernier scales are the three common types; each one reads to a smaller unit than the one before.

Plain scale

A plain scale has main divisions and one set of sub-divisions. It reads two units, e.g. metres and decimetres, but cannot read a third unit exactly. Example: RF = 1/200, so 1 cm = 2 m; a scale 10 cm long represents 20 m. Each main division (1 cm) is 2 m, and the left division is divided into 10 parts of 0.2 m each.

 m 2    1   0      2    4    6    8   (m)
   |....|...|------|----|----|----|----|
   <-sub-> <-- main divisions (2 m) -->

Reading: 7.4 m = 3 main divisions (6 m) + 7 sub-divisions on the left (1.4 m).

Diagonal scale

A diagonal scale reads three units (e.g. 100 m, 10 m, 1 m). Each sub-division is divided further by a set of equally spaced horizontal lines and diagonal lines drawn across them; the principle is that of similar triangles.

Example: RF = 1/4000 (1 cm = 40 m), to read 1 m up to 400 m. Length of scale = 400/4000 m = 10 cm. Divide it into 4 main divisions of 2.5 cm (100 m). Divide the left main division into 10 sub-divisions of 0.25 cm (10 m). Draw 10 equally spaced horizontal lines above the base. Join the top end of the first sub-division to the zero point on the base line, and draw diagonals parallel to it through the other sub-divisions. Each horizontal line moves the diagonal along by $\frac{1}{10}$ of 10 m = 1 m.

   10m
 10|  /./././././././/|
  8| / /  .. 
  6|/ / 
  4|  
  2|__|_____|_____|___|
   100 0   100  200 300

Measurement of 234 m: 200 m (2 main divisions) + 30 m (3 sub-divisions) + 4 m (4th horizontal line).

Vernier scale

A vernier scale is a small sliding scale used with a main scale. It reads a fraction of the smallest main division. If $n$ divisions of the vernier equal $(n-1)$ divisions of the main scale, then

$$\text{least count} = \frac{\text{value of 1 main division}}{n}$$

Example: 1 main division = 1 m and $n$ = 10, so the least count = 0.1 m. If the zero of the vernier is just past 4 m and the 3rd vernier line coincides with a main-scale line, the reading is $4 + 3 \times 0.1 = 4.3$ m.

The vernier is mostly used on instruments such as theodolites, target levelling staves and compasses, while plain and diagonal scales are drawn on maps.