Modular Arithmetic and the Vernier CaliperPaul Moulton

The Episcopal AcademyMorion, Pennsylvania 19066

If one looks into a silversmith’s tool box or into the set of toolsused by anyone who needs to measure things accurately, he is likelyto find there an F-shaped device known as a Vernier caliper. Withit the silversmith can measure the thickness of a sheet of silver ora machinist the diameter of a large shaft, both to an accuracy ofup to a thousandth of an inch. Yet, unlike most devices used tomake accurate measurements, the Vernier caliper performs this servicewithout demanding much in the way of careful handling or fussyproceedures.The Vernier scale was invented about three hundred fifty years

ago by Pierre Vernier, a French mathematician. By exploiting amathematical principle which we nowadays associate with modulararithmetic he was able to increase ten-fold the accuracy with whicha linear scale such as that used on a caliper can be read. His inventionis an excellent example of the way in which a little mathematicalingenuity can be used to increase man’s capacity to cope with thetasks otherwise beyond him�in this case his ability to perceive minutevariations in distance.For the sake of those who may be unfamiliar with this ingenious

device, we show a sketch of one being used to measure the diameterof a coin.

It will be noted that there are two scales, a fixed scale engravedon the main shaft ot the caliper, and, along the bottom of the window

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in the movable jaw, a short auxiliary scale. When the jaws are closed,the leftmost mark of the auxiliary scale points to 0 on the mainscale. Therefore, when the jaws are open, this left index points tothe number on the main scale that shows the distance between thejaws. The coin in the sketch is a little more than twenty-fourmillimeters.!

In fact, the coin is 24.3 mm, and to describe it as "a little morethan," does not do justice to the capability of the instrument. Byusing the auxiliary scale, it is possible to tell where in the intervalbetween twenty-four and twenty-five millimeters the measurementlies. Let us see how.

Reading the Scales

If we look closely at the two scales, which are shown in magnifiedform in Fig. 2, we note that the graduations of the auxiliary scaleare a little more closely spaced than are those of the scale aboveit. Furthermore, the graduations line up at only one point�the pointwhich is three spaces from the left on the auxiliary scale. It is bycounting these spaces to the left of the aligned marks that we knowthat the coin measures 24.3 mm.

FIG. 2

In order to appreciate the power of the auxiliary scale, it will bewell to consider some of the practicalities one must take into accountin constructing and reading an accurate linear scale. First of all, itis not the manufacturing techniques that limit us. Machines are readilyavailable that will engrave marks with a variation of a hundredthof a millimeter or less. It is the user of the scale who sets the limits.If the marks are to be seen and if they are to withstand handling,they must be reasonably wide and deep, at least a tenth of a millimeter.Then if they are to be numbered and counted, they must be widelyspaced, perhaps a half millimeter or more. The result is that evena very accurately engraved scale must have fairly broad, widely spacedcalibrations, and one cannot increase the accuracy with which such

1. For simplicity we shall confine our disucssion to a metric caliper. The principle of the Vernier scale canbe applied to other systems of measurement, including non-decimal ones.

Modular Arithmetic and the Vernier Caliper 457

a scale can be read by simply increasing the number of graduations.Thus it is possible to manufacture a caliper which is accurate to

a hundredth of a millimeter or better, but much of that accuracymust go to waste because the person who uses it cannot read itmore accurately than to the nearest half millimeter or so.

Vernier, however, found a way of circumventing this human limita-tion. By introducing his auxiliary scale he made it possible to splitthe scale reading process into two stages�one in which the readingis bracketed between two easily seen graduations on the main scaleand the other in which spaces are counted on an equally easy-to-seeauxiliary scale. By splitting the process into these two stages, hemade it humanly possible to read the caliper to the nearest tenthof a millimeter or better. It is rather gratifying to realize that thishas been accomplished not with magnifiers or other substantialadditions to the device, but with a little ingenuity.

"Why it Works"

But why does it work? Why does counting spaces which arethemselves about a millimeter wide give us a reading in tenths ofa millimeter? The answer to this question, it will turn out, is a ratherintriguing one because it ties modular arithmetic�which many peopleview as an impractical mathematical curiosity�in with a very practicaldevice. Of course since Vernier lived about two hundred years beforeGauss introduced the symbolism and formal study of modular arithme-tic, he cannot be imagined to have used its concepts as we nowknow them. Nevertheless the seeds of the idea were there. We cannow see the principle upon which his scale is based most easily interms of that branch of mathematics.To see this, let us suppose that we have a ten-hour clock with

the hours numbered from 0 to 9. Suppose, too, that we have somepoint, X, on that clock, and that we want to know exactly whereX is. If the hours were labeled, we could of course just look atthe label. If the clock were too small for such labels, however, wecould start at X and count the hours back to 0. Thus in Fig. 3,we could see that X is at 7 because that is the number of hoursback to 0.Modular arithmetic lets us do it another way, however, There is

a property of modular artihmetic which says that n - 1 and n 4-9 are the same number on a ten-hour clock. That is, moving ahead(clockwise) nine spaces is equivalent to moving back one. Insteadof counting individual hours back to 0, we could count the numberof nine-hour intervals ahead to 0. Thus, if we start at X in Fig.3 and count ahead by nines, it will take seven such counts, andwe know that Xis at 7. Fig. 4 shows two such counts.

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0

FIG. 3

But that might seem to be doing it the hard way. Each motionahead is nine times as far as its equivalent motion back. Actuallyit is the easy way. If the arcs are small and we need to count themindividually, it is easier to see and to count a big arc than a smallone. And if we have a pre-measured nine-hour interval, it is as easyto measure off a nine-hour interval as a one-hour interval. Both theseconditions are what we have in a Vernier scale.To translate our X-figuring process on the ten-hour clock to a Vernier

scale, let us do the following:Imagine that we have some point X lying between two of the

graduations, Yand Z, on a linear scale and that we want to determineto the nearest tenth where it is. To do this, let us construct a ten-hourclock whose circumference is exactly equal to the distance between

FIG. 4

Modular Arithmetic and the Vernier Caliper 459

the graduations on the linear scale. Then let us roll this clock alongthe scale with the 0 on the clock making contact with each of thelinear graduations. When the rolling clock reaches the X on the linearscale, let us make a corresponding mark Xon the clock.

Y X2

FIG. 5

Now let us roll the clock ahead through a series of nine-hour turns,starting at X, marking off the corresponding intervals on the linearscale. Each of these intervals will be nine tenths as long as the originallinear scale intervals, and each will be equivalent to counting backon the clock by a one-hour interval.

Y Z

T7FIG. 6

On the clock, we stop counting when the interval terminates at0. On the linear scale, we do the same thing: we count the numberof nine-hour intervals it takes until the end of one of the intervalscoincides (exactly or approximately) with one of the scale graduations.There will be as many of these nine-hour intervals as there are one-hourintervals on the clock from X back to 0. Each of these nine-hourintervals therefore corresponds with a tenth of an interval from Xback to Y. In counting them we have effectively counted the numberof tenths from X to Y.

Y Zl X i

FIG. 7

Of course, the process of constructing the necessary nine-hourintervals need be done only once. By marking off these intervalson a scale which one can slide along the main scale, we have only

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to slide this scale into place. And in the Vernier caliper, even theprocess of sliding the movable scale into place is accomplishedautomatically; it is engraved on the movable jaw itself. Thus to measurethe coin in Fig. 1, we had only to close the jaws on the coin, thenread the two scales.

Vernier’s invention is interesting on several counts. In additionto those already mentioned, it is interesting as an illustration of atype of mathematical thinking that is often overlooked. To most people,mathematical thinking consists of doing arithmetic, or perhaps provingthings. What is not recognized is that mathematics has many branches,each relating to reality in its own special way. Contemporary lifeabounds with inventions in which some fertile mind has recognizedthe utility of some supposedly sterile mathematical concept�inven-tions we often take for granted. The craftsman’s Vernier calipheris one of them.

REFERENCES

EVES, HOWARD. An Introduction to the History of Mathematics, Third Edition. NewYork: Holt, Rinehart and Winston, 1969.

"Measurement." Van Nostrand’s Scientific Encyclopedia, Fourth Edition."Vernier." The Columbia Encyclopedia, Third Edition.

LETTER TO THE EDITOR

The EditorSchool Science and Mathematics535 Kendall AvenueKalamazoo, Mich. 49007

Sir:

After reading "An Irrational Approach to the Number e" by Eiroy J.Bolduc, Jr. in your February 1976 Journal, I feel encouraged to submit ourmnemonic for TT.

With apoligies to Howard Eves, it goes as follows:Gee! I need a drink, alcoholic of course, after the class meetings.

Imaginary numbers aggravate for me the headache that occurs onMonday noon and may continue for an endless, agonizing trial.

This mnemonic is for the first 32 decimal places of TT, although only 31words appear after Gee. The 32nd decimal place luckily is a 0 which permittedus to stop.

Sincerely yours,Henry F. KahnMathematics Department HeadThomas A. Edison High SchoolPhiladelphia, PA 19133