## Theory of the Polar Planimeter.

### Vernier Units.

In the discussion of the Range of the Instrument it was shown that while the Planimeter could be made to record different number of Vernier Units for the same area by changing the length of the Tracer Arm, there was a maximum and minimum number of Vernier Units which could thus be obtained for any given area and any given instrument.

In the case of the particular Planimeter used, it was shown that for that instrument and for an actual area of 10,000 Sq. Millimeters, the instrument was capable of recording any desired number of Vernier Units between 907 and 2,721.

It very often happens that the conditions of some problem may demand that the instrument should deal with values greater or less than the limits thus found. This is very readily accomplished in effect by giving an assigned value to the actual Vernier Unit so as to cause one recorded actual unit to represent as many assigned units as the conditions of the problem may require.

Thus, suppose, with the instrument having the range already determined, we wish it to record say 3,200 Vernier Units for the given area.

As 2,721 is the maximum number of Vernier Units which this particular instrument is capable of recording for the given area, we can in effect obtain the desired result by assigning a value of 2 to each actual Unit recorded, and then making the Planimeter record 1,600 Actual Units for the given area.

Since one actual recorded Vernier Unit has an assigned or relative value of 2 for the given operation, by multiplying the number of actual Vernier Units record for the given operation by 2, the assigned or relative value, the result is exactly the same as though we had been able to make the instrument record the desired 3,200 units.

Hence in every problem the Vernier Units recorded during the operation of the Planimeter may have, and usually do have, two distinct values: one value the “Actual” or absolute value, and the other the assigned or “Relative” value dependent upon the special conditions of the given problem.

Thus, for example, in the case of the measurement of any area with the Planimeter, such for instance as a map drawn to some given scale, the number of recorded Vernier Units when multiplied by the “Actual” Vernier Unit value would give the actual area of the map traced in actual sq. ins. or other unit, while by multiplying the same by the assigned or “Relative Vernier Unit” value we would obtain the area in sq. feet, sq. miles, acres, or any other unit demanded by the scale of the map and other conditions of the problem.

As additional reason for the frequent use of the Relative Vernier Unit is that in most measurements with the Planimeter it is desirable that the length of tracer arm should be as great as possible consistent with other conditions, and this the employment of the Relative units will in most cases admit of.

The significance and use of the Relative Unit will be more clearly understood from its frequent use in the following descriptions of the practical applications of the Planimeter.