Norman F. Bray Herbert H. Lehman College Citv Universitv of New York Bronx, New York 10468
I I
The Vernier
I
T h e vernier was invented about 1630 by the French mathematician Pierre Vernier. It is probably best known to undergraduates in its use with some analytical balances, the vernier calipers, and the mercury barometer. Many texts describe how to use the vernier,' but the theory of its operation is virtually nonexistent. It appears that students and teachers alike accept the vernier on its merits and do not concern themselves with the reason why it works.
vernier will coincide exactly with one of the lines on the main scale. I n other words, it must be shown mathematically that the reading on the main scale of the corresponding vernier mark is some whole number multiple of the smallest scale division. By definition
Types of Verniers
where d is the smallest main scale division that is to be subdivided; yo is the fiducial, or index, mark on the vernier; and yl, y2, , , ., y, are the other vernier marks, 1, terms of yo, which is the actual main scale reading
If a scale is to be read to one Nth of the smallest divi1 sion, the auxiliary scale, or vernier, will have N evenly spaced lines equal to N lines on the main scale. The "ernier having N-+ 1auxiliary lines is known as a direct vernier,%and one auxiliary division equals ( N - 1 ) / N scale divisions; the vernier with N - 1 auxiliary lines is known as a retrograde vernier2 and has (N 1 ) / N scale divisions per vernier division. Those = most commonly used are direct verniers with and N = 10. The theory of operation for these two special cases is given below.
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+
Theory of Operation
I n general, it is necessary to Prove that, with the exception of yo and y ~ one , ~and only one line on the
1 vernier division = (N 7 1)/N scale divisions
(1)
and (YL - YO)
=
d(N 7 1)lN
(2)
D, P,, A N D GARLAND, C, W,r crExperiments in physical Chemistw," (2nd Ed.), McGraw-Hill Book Co., New ~ o r k 1967, , p. 431. SKOOG, D. A., AND WEST,D. M., "Fundamentals of Analytical Chemistry," Holt, Rinehart, and Winston, H. H., FURMAN, N. H., A N D New York, 1963, p. 85. WILLARD, BRICKER,C. E., "Elements of Quantitative Analysis: Theory and Practice," (4th Ed.), D. Van Nostrand Co., Inc., Princeton, N. J,. 19sa, D. 40. 1 &ous;,'w. H., (Editor), "McGraw-Hill Encyclopedia of Science and Technology," McGraw-Hill Book Co., New York, 1960, Val. 14, p. 306. yo and y~ are equivalent positions; when one is coincident with a line on the main scale, the other will also he coincident with a line on the main scale ( N T 1 ) units away.
Volume 47, Number 1 , January 1970
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75
Table 1.
Table 2. Vernier Line Readings Direct Vernier, Special Cases
Vernier Line Readings as Function of Fiducial Mark, yoa
-
+
= ~ / o id(N F 1 ) / N i = 0 , 1 , 2 ,..., N
~6
a
a Negative values are for a. direct vernier, positive values for retrograde vernier.
YI = d ( N F 1 ) l N
+ YO
(3)
Since all vernier divisions are equal (?4%- yo) = (yr - Yl) (Y*
+ (YL- yo)
- YO)
(4)
- yo) = 2d(N F 1 ) I N
and y2 = 2d(N ? 1 ) / N
(5)
+ Yo
(6) Special Case:
Then (Y3
= 2(Y1
- Y O )=
(Y3 - Y * )
+ (Y2 - Yd +(Y, - Y O ) -
( ~ r yo)
=
= 3(Y,
- YO)
(7)
3d(N F 1 ) l N
(8)
+ YO
(9)
and ya = 3d(N F 1 ) l N
The remaining values of yr are obtained in a similar manner, and some of these are listed in Table 1. I n general ~i
where i
est tenth) will determine which of the y, values will be integral and which vernier line will coincide with a line on the main scale. That one value will be the same as the last digit of the yo value. For example, if the yo reading is 32.7 as in the figure, the other values of ye are determined from eqn. (10) and are given in Table 2. As can be seen, only yr is an integral value.
=
= id(N ? 1 ) l N
+
YO
(10)
0, 1, 2, . . . , N .
Special Case:
Direct, N = 10, d = 1
This type of vernier is used on most mercury barometers, the vernier calipers, Chain-o-Rlatic balances, and some single pan substitution-type balance^.^ These balances will be accurate to 1mg without the use of the vernier, while the vernier allows the reading of the scale to 0.1 mg.S This vernier divides each main scale division (1 mg) into 10 equal parts. The values of y, are given in Table 2. It can be seen 50-4 that the additive term ends in a different integer for each y~value,and therefore, One and onevalue of yt will be some whole number multiple of the smallest 30 scale division; in this case, it will be any integralvalue ending in a number between 20 and nine' Thus' the value of yo on the main Direct verniec N = 10, d = 1. Reading ir 32.7. scale (rounded to the near-
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76 / Journal o f Chemical Education
Direct, N = 5, d = 0.5
A vernier of this type is used on the projection reading device of the Christian Becker AB-1 balance, and it divides each main scale division (0.5 mg) into five equal parts. The values of y, are given in Table 2. For a given y, to coincide with a line on the main scale, its value must end in either zero or five. Inspection will show that one and only one line on the vernier will end in these values when the last digit of yo (rounded to the nearest 0.1 mg) ends in any number between zero and five. Conclusion
As eqn. (10) shows no dependence of yr on a specific position of the fiducial mark, the operation of the vernier is independent of its location on the main scale and may be used over the entire range of main scale values. It should be obvious that the result is independent of the position of the decimal point. Verniers for which N is neither five nor ten are known and in use. A treatment similar to the one given here may be made for any value of N in order to test the validity of any specific application. Acknowledgment
The author is deeply indebted to Diana C. Bray for reading the manuscript and making many helpful snggestions. Examples of single pan balances which use this type of vernier are the Ainsworth RighbA-Weigh, Mettler H5, and
M%$i2~i is said concerning the precision or reproducibility of the m e a m m e n t here; i t is assumed that the belance is a type capable of the stated accuracy.
