a scale excellently adapted for the graduation of mathematical instruments, thus called from its inventor Peter Vernier, a person of distinction in the Franche Comté. See NONIUS.
Vernier's method is derived from the following principle. If two equal right lines, or circular arcs, A, B, are so divided, that the number of equal divisions in B is one less than the number of equal divisions of A, then will the excess of one division of B above one division of A be compounded of the ratios of one of A to A, and of one of B to B.
For let A contain 11 parts, then one of A to A is as 1 to 11, or \( \frac{1}{11} \). Let B contain 10 parts, then one of B to B is as 1 to 10, or \( \frac{1}{10} \). Now \( \frac{1}{10} - \frac{1}{11} = \frac{11-10}{10 \times 11} = \frac{1}{10 \times 11} = \frac{1}{10} \times \frac{1}{11} \).
Or if B contains n parts, and A contains n+1 parts; then \( \frac{1}{n} \) is one part of B, and \( \frac{1}{n+1} \) is one part of A.
And \( \frac{1}{n} - \frac{1}{n+1} = \frac{n+1-n}{n \times n+1} = \frac{1}{n} \times \frac{1}{n+1} \).
The most commodious divisions, and their aliquot parts, into which the degrees on the circular limb of an instrument may be supposed to be divided, depend on the radius of that instrument.
Let R be the radius of a circle in inches; and a degree to be divided into n parts, each being \( \frac{1}{p} \)-th part of an inch.
Now the circumference of a circle, in parts of its diameter 2R inches, is \( 3,1415926 \times 2R :: 10 ; \frac{3,1415926}{360} \times 2R \) inches.
Or, 0,01745329×R is the length of one degree in inches.
Or, 0,01745329×R×p is the length of 1°, in pth parts of an inch.
But as every degree contains n times such parts, therefore \( n = 0,01745329 \times R \times p \).
The most commodious perceptible division is \( \frac{1}{8} \) or \( \frac{1}{10} \) of an inch.
Example. Suppose an instrument of 30 inches radius, into how many convenient parts may each degree be divided? how many of these parts are to go to the breadth of the vernier, and to what parts of a degree may an observation be made by that instrument?
Now 0,01745×R=0,5236 inches, the length of each degree: and if p be supposed about \( \frac{1}{8} \) of an inch for one division; then 0,5236×p=4,188 shows the number of such parts in a degree. But as this number must be an integer, let it be 4, each being 1.5°: and let the breadth of the vernier contain 31 of those parts, or \( \frac{7}{8} \), and be divided into 30 parts.
Here \( n = \frac{1}{4} ; m = \frac{1}{30} \); then \( \frac{1}{4} \times \frac{1}{30} = \frac{1}{120} \) of a degree, gree, or 30', which is the least part of a degree that instrument can show.
If \( n = \frac{1}{5} \) and \( m = \frac{1}{36} \); then \( \frac{1}{5} \times \frac{1}{36} = \frac{60}{5 \times 36} \) of a minute, or 20''.
The following table, taken as examples in the instruments commonly made from 3 inches to 8 feet radius, shows the divisions of the limb to nearest tenths of inches, so as to be an aliquot of 60's, and what parts of a degree may be estimated by the vernier, it being divided into such equal parts, and containing such degrees as their columns show.
<table> <tr> <th>Rad. inches.</th> <th>Parts of a degree.</th> <th>Parts in vernier.</th> <th>Breadth of vernier</th> <th>Parts observed.</th> </tr> <tr><td>3</td><td>1</td><td>15</td><td>1 5/7</td><td>4' 9''</td></tr> <tr><td>6</td><td>1</td><td>20</td><td>2 0/7</td><td>3 0</td></tr> <tr><td>9</td><td>2</td><td>20</td><td>1 0/7</td><td>1 30</td></tr> <tr><td>12</td><td>2</td><td>24</td><td>1 2/7</td><td>1 15</td></tr> <tr><td>15</td><td>3</td><td>20</td><td>0 3/7</td><td>1 0</td></tr> <tr><td>18</td><td>3</td><td>30</td><td>1 0/7</td><td>0 40</td></tr> <tr><td>21</td><td>4</td><td>30</td><td>7/7</td><td>0 30</td></tr> <tr><td>24</td><td>4</td><td>36</td><td>9/7</td><td>0 25</td></tr> <tr><td>30</td><td>5</td><td>30</td><td>7/7</td><td>0 20</td></tr> <tr><td>36</td><td>6</td><td>30</td><td>5/7</td><td>0 20</td></tr> <tr><td>42</td><td>8</td><td>30</td><td>3/7</td><td>0 15</td></tr> <tr><td>48</td><td>9</td><td>40</td><td>4/7</td><td>0 10</td></tr> <tr><td>60</td><td>10</td><td>30</td><td>3 1/7</td><td>0 10</td></tr> <tr><td>72</td><td>12</td><td>30</td><td>2 1/7</td><td>0 10</td></tr> <tr><td>84</td><td>15</td><td>40</td><td>2 7/7</td><td>0 6</td></tr> <tr><td>96</td><td>15</td><td>60</td><td>4</td><td>0 4</td></tr> </table>
By altering the number of divisions, either in the degrees or in the vernier, or in both, an angle can be observed to a different degree of accuracy. Thus, to a radius of 30 inches, if a degree be divided into 12 parts, each being five minutes, and the breadth of the vernier be 21 such parts, or 1 3/7, and divided into 20 parts, then \( \frac{1}{12} \times \frac{1}{20} = \frac{1}{240} = 15'' \): or taking the breadth of the vernier 2 7/7, and divided into 30 parts; then \( \frac{1}{12} \times \frac{1}{30} = \frac{1}{360} \), or 10'': Or \( \frac{1}{12} \times \frac{1}{50} = \frac{1}{600} = 6'' \); where the breadth of the vernier is 4 2/7.