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CHAPTER II.

The Polar Planimeter.

Point of Beginning.

The staring point of the Tracing should be at or near the point as shown on the diagram. This point is the intersection of the “Constant Circle” or “Base” with the outline of the figure to be traced and is selected as the most favorable position for both beginning and ending the operation of tracing owing to the fact that it is at this point of the outline there is the least amount of movement of the wheel for any given movement of the Tracer, and hence at this point that any accidental error or deviation of the Tracer will have its minimum effect on the movement of the Wheel.

A further consideration in selecting the best position of the Planimeter with respect to any area to be traced is that the Instrument should be so placed that in tracing its outline no important side of the outline shall be parallel to or near the “Base.” The reason for this is evident from the conditions named above as governing the choice of a starting point for the tracing.

As has already been stated the description of the Polar Planimeter just given will apply in all essential particulars to all forms of the Polar type of Planimeter, as the theory and operation are identical in all.

In the higher types of the instrument such as the Rolling Spherical, the Precision Disk Planimeter, the Spherical Polar, the Integraphs and others of like character— some of the more important of which are described later in our discussion— other principles are involved in construction and theory which admit of the performing of a class of mathematical operation impossible to the simple Polar form. But in operation possible to both Precision and Polar types the more complicated nature of the construction and theory of the so-called Precision instruments and the additional precautions taken in their design and operation to avoid every possible source of error must necessarily permit of the obtaining of results by their use of a relatively higher degree of accuracy than is obtainable by the simpler form. While this may be of advantage in the case of a small class of problems necessitating an unusual degree of accuracy in the instrument, in the great majority of cases if the adjustments as described above are carefully made and the conditions for accuracy as given are closely observed, the Polar Planimeter will give results of such a degree of accuracy as to be far within the error allowable in any given problem and far higher than would be possible by any other known means.

In some forms the Polar Planimeter either by change in some minor details of mechanical construction or by a special attachment the instrument is made especially adaptable to some one particular operation which by this means it is enabled to perform with greater facility without lessening its value for other operations. These special forms will be described in their proper place when describing the use of the Planimeter in its practical application involving the operation in question, as will also further notes on the adjustments, methods of operating and conditions requisite for maximum efficiency in the use of the instrument.

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