## Measurement of Plane Areas.

### Use of the Constant.

It has already been stated that the instrument when measuring a figure with the Pole on the interior of the figure records only the Reading due to the difference in area between the area of the given figure and the area of the Constant Circle for the Setting to which the Planimeter is adjusted. It has also been explained that when the area of the given figure is less than the area of the Constant Circle for the given Setting the resultant rolling of the wheel will be backwards and the Reading will be a – one. If the area of the given figure be greater than the area of the Constant the resultant rolling of the wheel will be a forward or positive rolling and the sign of the reading will be +. The final Reading being, as has been said, the Reading due to an area which is the difference in area between the area of the figure traced and the Constant Circle must then be added algebraically to the area of the Constant Circle in order to get the total Reading for the figure traced.

If the resultant Reading is found to be a backward or – one, it shows that the area of the given figure is less than the area of the Constant Circle and adding a Constant to a – Reading is equivalent to subtracting the Reading from the Constant: This is to be expected since the Reading, being the Reading for the difference between the areas of figure and constant circle, when subtracted from the Constant (which is really the Reading for the Constant Circle when traced with an exterior pole) evidently gives the Reading for the given figure.

And conversely, when the resultant Reading is +, we add a Constant to a + Reading to get the reading of the given figure, since the area of the figure is greater than the area of the Constant Circle.

In our example, let us suppose that the final Reading after tracing the given figure to be say 8,743, which we know from having kept watch of the wheel is a backward or – movement. The Reading will then evidently be 10,000 – 8,743 = 1,257, which as we have seen is –. Looking in our Table we find the value of the Constant to be 18,355. Adding these quantities we have 18,355 + (–1257) = 18,355 – 1257 = 17,098 as the total Reading for the entire figure. Multiplying this be 20.0 the value of the Rel. Vernier Unit, we have for the area of the figure, 17,098 × 20.0 = 341,960 Sq. Ft. which is the area required.

In this example if our final Reading had been say 1,257 and we know by observation that it was a forward or positive relation we should have had

18,355 + (+1257) = 18,355 + 1257 = 19,612
and 19,612 × 20.0 would be the required area = 392,240 Sq. Ft.

The description of this method of Planimeter measurement just given will be more clearly understood at least as to the “why and wherefore” from a demonstration and discussion of the theoretical principles involved in the construction and operation of the Polar Planimeter which are given in Chapter III. The derivation and significance of the Constant are there fully explained but it has not been thought best to introduce further theoretical explanation here from fear of complicating the description just given for the practical use of the Planimeter.

The intent of the descriptions given throughout of the use of the Instrument, both of the methods of measurement already given and in the description of the use of the Planimeter in all its practical application, has been to make them so clear so as to admit of the intelligent use of the instrument in every operation without reference to the theoretical conditions involved.