VERNIER SCALE

VERNIER SCALE, 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 $\backslash frac\{1\}\{11\}$. Let B contain 10 parts, then one of B to

B is as 1 to 10, or $\backslash frac\{1\}\{10\}$. Now $\backslash frac\{1\}\{10\}\; -\; \backslash frac\{1\}\{11\}\; =\; \backslash frac\{11-10\}\{10\; \backslash times\; 11\}\; =\; \backslash frac\{1\}\{10\; \backslash times\; 11\}\; =\; \backslash frac\{1\}\{10\}\; \backslash times\; \backslash frac\{1\}\{11\}$.

Or if B contains $n$ parts, and A contains $n+1$ parts; then $\backslash frac\{1\}\{n\}$ is one part of B, and $\backslash frac\{1\}\{n+1\}$ is one part of A. And $\backslash frac\{1\}\{n\}\; -\; \backslash frac\{1\}\{n+1\}\; =\; \backslash frac\{n+1-n\}\{n\; \backslash times\; n+1\}\; =\; \backslash frac\{1\}\{n\}\; \backslash times\; \backslash frac\{1\}\{n+1\}$.

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 $\backslash frac\{1\}\{n\}$th part of an inch.

Now the circumference of a circle, in parts of its diameter 2 R inches, is $3,1415926\; \backslash times\; 2\; R$ inches.

Then $360^\backslash circ\; :\; 3,1415926\; \backslash times\; 2\; R\; ::\; 1^\backslash circ\; :\; \backslash frac\{3,1415926\}\{360\}\; \backslash times\; 2\; R$ inches.

Or, $0,01745329\; \backslash times\; R$ is the length of one degree in inches.

Or, $0,01745329\; \backslash times\; R\; \backslash times\; p$ is the length of $1^\backslash circ$, in $p$th parts of an inch.

But as every degree contains $n$ times such parts, therefore $n\; =\; 0,01745329\; \backslash times\; R\; \backslash times\; p$.

The most commodious perceptible division is $\backslash frac\{1\}\{8\}$ or $\backslash frac\{1\}\{10\}$ 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 $0,01745\; \backslash times\; R\; =\; 0,5236$ inches, the length of each degree: and if $p$ be supposed about $\backslash frac\{1\}\{8\}$ of an inch for one division; then $0,5236\; \backslash times\; p\; =\; 4,188$ shows the number of such parts in a degree. But as this number must be an integer, let it be 4, each being $15\text{'}$; and let the breadth of the vernier contain 31 of those parts, or $7\backslash frac\{1\}\{2\}$, and be divided into 30 parts.

Here $n\; =\; \backslash frac\{1\}\{4\}$; $m\; =\; \backslash frac\{1\}\{30\}$; then $\backslash frac\{1\}\{4\}\; \backslash times\; \backslash frac\{1\}\{30\}\; =\; \backslash frac\{1\}\{120\}$ of a de-

gree, or $30\text{'}$, which is the least part of a degree that instrument can show.

If $n\; =\; \backslash frac\{1\}\{5\}$, and $m\; =\; \backslash frac\{1\}\{36\}$; then $\backslash frac\{1\}\{5\}\; \backslash times\; \backslash frac\{1\}\{36\}\; =\; \backslash frac\{60\}\{5\; \backslash times\; 36\}$ of a minute, or $20\text{'}\text{'}$.

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 $60^\backslash circ$'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.

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 $1\backslash frac\{1\}\{2\}^\backslash circ$, and divided into 20 parts, then $\backslash frac\{1\}\{12\}\; \backslash times\; \backslash frac\{1\}\{20\}\; =$

$\backslash frac\{1^\backslash circ\}\{240\}\; =\; 15\text{'}\text{'}$; or taking the breadth of the vernier $2\backslash frac\{1\}\{2\}^\backslash circ$, and divided into 30 parts; then $\backslash frac\{1\}\{12\}\; \backslash times\; \backslash frac\{1\}\{30\}\; =\; \backslash frac\{1^\backslash circ\}\{360\}$, or $10\text{'}\text{'}$:

Or $\backslash frac\{1\}\{12\}\; \backslash times\; \backslash frac\{1\}\{50\}\; =\; \backslash frac\{1^\backslash circ\}\{600\}\; =\; 6\text{'}\text{'}$; where the breadth of the vernier is $4\backslash frac\{1\}\{2\}^\backslash circ$.