VERNIER, a scale adapted for the gra duation of mathematical instruments, so called from Pierre Vernier the inventor. Under the article BAROMETER will be seen some account of this scale as applied to that instrument; here we shall take it up more generally.
This scale is derived from the following principle. If two equal right lines, or circular arcs, A 13, are so divided, '.nat the number of equal divisions in B is one less than the number of equal divisions in A then will the excess of 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 1 to 11, or Let B contain 10 parts, the n one of B to B is as 1 to 10, or Now 10 1 1 1 1 2U 11 x Or if B contains n parts, and a contains ną1 parts ; then- is one part of B, and 1 1 1 n+1 .------ is one part of A. And - = n+n n n + 1 n 1 n x n +1 n f ' The most commodious divisions, and their aliquot parts, into which the degrees n-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 th part of an inch Now the circumference of a circle, in parts of its diameter 2 R. inches, is 3.1415 926 X 2 R inches.
Then 360° : 3.1415926 X 2 R : : 1° : 3.1415926 300 X 2 R inches.
Or, 0.01745329 X R is the length of ope degree in inches.
Or, 0.01745329 x R X p is the length of 1°, in pth parts of an inch.
lent as every degree contains n times such parts, therefore n = 0.01745329 X R X p.
The most commodious perceptible di 1 1 vision is 5 or of an inch.
Example. Suppose an instrument of 30 inches radius, into how raw 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 X R = 0.5236 inches, the length of each degree : and if p is 1 supposed about 8 - of an inch for one divi sion ; then 0.5236 X 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' ; and let the breadth of the vernier contain 31 of those parts, or 7.2°, and be divided into 30 parts.
1 1 1 1 Here n = 4 ; = 120 of a degree, or 30', which is the least part of a degree that instrument can show.
1 ' 36 1 1 If n = and m ; then 5 -x= Y 36 60 of a minute, or 20".
5 X S6