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Measurement of Plane Areas.

II. Measurement of Small Areas.

Example of Measurement.

Example 1.   Let us suppose the figure whose area is desired to be drawn to a scale of 1 inch = 30 feet, and we desire to measure the area with the Polar Planimeter— the area to be expressed in Sq. Ft.

Looking in Table 1 we find the Setting of the scale 1" = 30' to be 121.8. Unscrewing the screw L (Plate 1, Fig. 1), we slide the Tracer Arm RS through the sleeves of the carriage until the first small division beyond the division line marked 12 is made to coincide with the Zero line of the Carriage Vernier V. Clamping the screw L the arm is moved by the slow motion screw M until the small line 8 on the Carriage Vernier is made to coincide with the first line encountered by it on the Tracer Arm which in this case is the line 129. This being accomplished the Planimeter is now in adjustment for making the desired measurement.

The Planimeter is now placed in correct position with respect to the figure to be measured, and the Pole placed in the “most favorable position,” as explained above, and the Tracer brought to the place of beginning or starting point determined upon. The Reading of the Instrument is then taken; let us suppose that the index of the Counting Wheel O points to or a little beyond the line marked 3, and that the graduated drum of the wheel W reads 324. The first Reading of the Planimeter is then 3324.

The Tracer is now carefully guided along the periphery of the given figure in the manner and with regard to the conditions already given until the Tracer having traced the complete boundary of the figure arrives again at the point of beginning. The Reading of the Planimeter is again taken, and let us suppose that the Counting Wheel now reads 7 and the wheel W reads 597, which makes the reading of the instrument 7597. If now we subtract the first Reading 3324, from the second or final Reading 7597, we have for the Reading for the given figure, 7597 – 3324 = 4273, which shows that the Integrating Wheel W has made 4 whole revolutions and 273 thousandths of a revolution while the Tracer has been tracing the periphery of the given figure; or which is the same thing, the wheel has recorded 4273 Vernier Units while tracing the given figure.

Looking now in Table 1 for the value of the Relative Vernier Units we find that for the given scale, 1in. = 30 ft., one Vernier Unit recorded by the wheel represents an area of 10 sq. ft., and as the wheel has recorded 4273 Vernier Units for the given figure the area of that figure must be 4273 × 10 = 42730 Sq. Ft., which is the required result.

If for purpose of checking the accuracy or for other reasons the actual area in square inches of the figure just measured should be desired it can of course be readily measured by setting the Planimeter to the Setting 145.3 which with Vernier Unit .01 is the Setting for the natural scale of 1 inch = 1 inch for the instrument used. The figure being again traced with the new Setting will then give a Reading which when multiplied by .01, the value of the Vernier Unit, will give the actual area in square inches.

To facilitate this operation and admit of the measurement of the given area in square inches without changing the Setting or having to make a second tracing of the figure, a set of factors have been calculated and places in all the Tables giving what we have termed the Actual Vernier Unit for each of the given scales or settings. The value of the Actual Vernier Unit in any case is such that if we multiply the Reading for any figure traced by the value of the Actual Vernier Unit found in the proper column opposite the scale in question, the product will be the actual area of the given figure expressed in square inches. For example, in the example just given, the reading of the instrument for the given figure was, as we have seen, 4273 Wheel Units. Looking in the column headed Actual Vernier Units opposite the given scale, 1" = 30' we find the value of the Actual Unit to be .0111, which means that each Unit recorded represents an actual area of .0111 Sq. Ins. The actual Area of the entire figure will be 4273 × .0111 = 47.47 Sq. Ins. Since in our example 1 Sq. In. = 900 Sq. Ft. the check is made by simply multiplying the total actual Area of 47.47 Sq. Ins. by 900, which gives the 42730 Sq. Ft. found by the first measurement.

In the example just given should the area be desired in Acres instead of Sq. Ft. the measurement would be made in exactly the same manner, except that the Planimeter would be set to 50.0, which is the Setting given in Table 5 for Scale 1 in. = 30 ft., and the resultant Reading would be multiplied by .0003, which is the corresponding value of the Relative Vernier Unit for the scale in question.

When we consider that the area of any plane figure, regardless of the irregularity or nature of its periphery which may be curved or broken to any extent, can be measured by the simple operation described, and its area determined within a degree of accuracy of say 1/20 of 1 per cent. by the simple Polar Planimeter, we get some idea of the almost incalculable value of the instrument as an aid to professional or scientific work.

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