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Measurement of Plane Areas.

I. Measurement in General.

Methods of Measurement.

There are two methods by which the Area of any plane figure may be accurately measured by the Polar Planimeter, the method selected for any given case being dependent on the size of the given Area and capacity of the Planimeter used. The first of these two methods which we are about to describe is given under the heading “Measurement of Small Areas,” and is the method invariably employed when the size and shape of the figure to be measured is such as to allow of every point of its perimeter being reached by the Tracer without moving or shifting the position of the Pole during the tracing, and having the Pole outside of the given area.

The second method of measurements is described under the heading “Measurement of Large Areas,” and is used when the area of the given figure is too large to be measured by the first method, or when the size of shape of the figure to be traced is such as not to allow of every point on its perimeter being reached by the Tracer when tracing the figure without moving the Pole— the Pole being on the outside of the given figure.

These two methods are distinguished and usually designated as “Measuring with Exterior Pole” is the method always selected when the conditions as specified above admit of its employment.

The measurement of the area of plane figures will now be taken up, the use of the Planimeter in both methods of measurement described in detail, and the conditions, both theoretical and practical, on which the maximum degree of accuracy in operating and results depend, will be explained in as clear a manner as possible.

As the actual operation of the Polar Planimeter in all its many particular applications is exactly the same as in the single one of obtaining the area of a plane figure, the following description of the manner of operating and the directions given for obtaining the highest possible degree of accuracy and efficiency will apply with equal force to its use in every application, and should be clearly understood and carefully followed in every case.

Keeping in mind what we have elsewhere termed the “General Equation” or Fundamental Principle of the Polar Planimeter which was deduced in the demonstration of the General Theory of the Planimeter given in Chapter III, the theoretical conditions involved in each and every operation of the instrument can be clearly seen. It will also prove how essential to accurate results it is that the directions given be implicitly followed, as well as give an intelligent understanding of the working of the Planimeter, and an appreciation of the wonderful accuracy and practical value of the Planimeter in every form of Engineering and Scientific computation.

The measurement of the area of any plane figure by means of the Polar Planimeter is a very simple operation, and when carefully performed with a Planimeter in good condition and adjustment and with the conditions for accuracy properly observed and complied with, will give results with a degree of accuracy not attainable by any other method of measurement.

This is especially true in the case of figures having irregular bounding lines, or when a portion or all of the periphery of the given area is composed of curved or broken lines, since the shape or nature of the outlines of the figure have no influence whatever on the accuracy of the measurement.

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