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The Schnöckel compensation planimeter by Dr N. Wolff, Cologne

Among the tools for calculating graphical areas, planimeters based on the Coradi system occupy first place. Where the accuracy of the calculation is concerned, they have not been equaled by any other instrument, but their relatively high price has prevented them from being used widely, which one would wish for in the interest of the rational use of labor power.

Schnöckel [1] enters the scene with a new instrument. The basic theory is the same as for Prytz's bar planimeter, but the error of the latter is reduced to a minimum by design innovations, which make the instrument suitable for more precise calculations. The principles of Prytz's rod planimeter are therefore briefly mentioned at the beginning of this treatise.

The instrument [2] consists of a rod of length l with a tracing point at one end and the other end bent into a wedge-shaped cutting edge. The tracing point is moved round the contour of the figure to be evaluated. The area is equal to the product of the rod length and the distance a between the starting and end positions of the cutting edge.

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In Figure 1, F is the tracing point, s is the cutting edge. When the tracing point F moves clockwise round the figure, the cutting edge travels the path S-A-B-C-S'. The area of the figure J is equal to a.l, where l is the rod length and a means the straight connecting line between S and S'. However, J = a.l is only a crude approximation of the area. A residual quantity, the size of which depends on various circumstances, including the position of the tracing arm and the size and shape of the figure, is excluded. For details, please refer to Jordan, Handb. d.Verm., .Bd II, 8th edition, p.147 f.f and Hamann, Z.f.Verm., 1896, p.643 f.f.

The residual quantity naturally has a major influence on the accuracy of the calculation. Various authors have suggested ways to diminish this residual quantity. But it is only Schnöckel who has managed to make the rod planimeter usable for more precise calculations through special design features.

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The Schnöckel planimeter (Fig. 2) consists of an approximately 27 cm long metal rod a, which rests on a ball k. A transparent celluloid plate b with a tracing mark f and a tracing line g is attached to one end, and a ball stop i is attached to the other end. An index line is inscribed in the beveled surface of the ball stop, which, in conjunction with a scale c, allows the respective positions of the rod to be read. The only thing the instrument has in common with the original Prytz rod planimeter is the rod: the tracing point is replaced by the tracing mark, the cutting edge is replaced by the ball and index line. Movements in the rod direction and rotating movements are smoother and more even due to the ball. The distance between the ball and the tracing mark does not remain constant, like the tracing pin and cutting edge of the Prytz instrument, but changes constantly as the instrument moves around.

When in use, the rod is placed on the ball, which rolls in a groove as the contour is traced. The rod is placed over the figure to be evaluated so that its axis divides the figure into two approximately equal halves, and it points in the direction of the largest diameter for elongated figures. The ball, at the ball start mark, should be to the left of the operator, and the celluloid plate with tracing mark should be to the right. The tracing mark f is brought over a corner point of the figure or at the end of a short line l. (Fig 2.) The scale is aligned with the index line, preferably at 0 or 1.

You press the pin n in one corner of the ruler lightly into the paper and carefully turn the ruler to the left out of the area of the moving rod. Trace round the figure to the right. After the tracing mark has been returned to the starting point, you rotate the scale back to the index mark and take a second reading. The reading must be multiplied by the corresponding constant to obtain the area. According to the inventor, one circuit should be sufficient for figures of more than half a square meter, while smaller figures are traced twice in one go. To increase the reading accuracy, the inventor also recommends using the micrometer scale m inscribed on the celluloid plate (Fig. 2). If only the index line is used for reading, the tens are estimated. So an estimation error of 10 units can occur that is insignificant for larger areas, but significant in smaller areas. The inventor seeks to remedy this deficiency with the above-mentioned micrometer scale.

In the author's opinion, using the micrometer scale is too cumbersome. It is one of the few disadvantages of this otherwise handy and, as should be noted, precise instrument. It would be more logical to combine the micrometer scale with the pointer and to install a vernier instead of the simple pointer. The inventor considers the lack of a vernier to be one of the advantages of the instrument, because it is difficult to teach less-trained assistants how to use the vernier.

The latter argument may not be entirely sufficient to justify the lack of a vernier. When making an instrument, the ultimate goal is to achieve the greatest possible accuracy combined with the greatest possible handiness. The users of the instrument must adapt to this. Also, at least in practice, one does not allow untrained people to carry out calculations, but rather those who have already had some practice. A hired assistant will learn how to use the vernier in a very short time. In addition, for larger areas, using just the 0-line of the vernier can be left to the operator.

If it is not possible to make a double circuit in one go for larger areas, a different position of the driving rod should be chosen for the second circuit, preferably one perpendicular to the first one. However, with elongated figures, the requirement that the rod remains in the direction of the largest diameter must be met. In this case, the start and travel point are swapped on the second circuit by turning the travel rod by 180°.

If the documents are poor, the results can be improved by tracing first clockwise and then anti-clockwise. The inventor calls this “returning the tracing point”. This process is explained using an example taken from the inventor's instructions. After setting the tracing mark, you can read off the scale 0064. After the clockwise circuit 2376, the counter-clockwise achieves 0069. The result is as follows:

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As with other area measuring tools, the accuracy of the results depends on various circumstances. The size and shape of the figure, the nature of the surface, the position of the arm, all of these are factors that influence accuracy. In his instructions for practical use, the inventor has laid down the principles for achieving the greatest possible accuracy; please refer to these instructions.

In the following calculations, figures of different shapes (circle, square, rectangle, irregular shape) were measured by tracing around them twice in one go. For the larger figures, where the scale length was too short, two different rod positions were chosen. The calculations were carried out on rough and smooth surfaces as well as on old, wrinkled drawing paper. In view of the fact that the investigations are intended to allow an assessment of the extent to which the instrument can be used in calculations in everyday practice, the tracing was carried out with the care that is usually applied in practice. The micrometer scale was not used for reasons already mentioned. Areas smaller than 600 sq. mm were not calculated because the estimation error is too obvious for these.

In order not to have to take the paper input into account, the figures were calculated immediately after application. The first part of the investigation is only intended to provide a general overview of the size of the errors that arise. In the second part, a 10 cm square was circled ten times with the instrument in different positions in order to be able to determine the internal error and the overall error of the instrument.

Before discussing the results, the following should be pointed out: when determining the nominal area of the figures, the error of the drawing was disregarded; particular care was taken when making the figures. The contents of the irregular figures were determined using a compass and ruler to obtain the nominal sum.

While the surface influences the accuracy of the Coradi polar planimeter, it plays an insignificant role in this instrument. Calculations made on wrinkled 100-year-old maps gave just as good a result as those made on smooth paper. This shows the superiority of the Schnöckel planimeter, which slides easily on the ball, compared to the compensation planimeter, which is more strongly influenced by external circumstances. There is a mismatch between the two instruments when tracing around elongated figures. The error grows to such an extent that in practice one will abandon the instrument in favour of the harp planimeter or similar aids. This in no way diminishes the value of the instrument, as it is a defect that is inherent in more or less all contour tracing instruments.

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To determine the internal and total error, the following definition is provided. Internal error should be understood as the deviation from the arithmetic mean that occurs when the same figure is traced around several times, while the overall error should also include the instrument errors. The first is obtained by the formula

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The true errors e may be used to determine the total error. True, the true errors cannot be obtained with any rigor, since it is not possible to draw a figure mathematically precisely. However, the errors in the graphic representation are less important than the other errors. The overall error obtained in this way can provide better information about the usability of the instrument than the internal error, which only has theoretical significance. The tests resulted in an internal error μ = ± 0.41% and a total error of ± 0.81% for an area of 100 square cm. Jordan's error function is:

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Jordan chooses the value 0.03 for the constant k, taking the total errors into account. Assuming Jordan's values, the average error is ± 0.3% for an area of 100 square cm. However, it is not specified which planimeter this error should apply to.

According to the inventor's investigations, the error formula μ = ± 0.2√F applies to the present instrument. For an area of 100 square cm, this results in an error of ± 0.2%. The author could not achieve this level of accuracy with the instrument.

In order to determine whether the present instrument can also be used for more precise calculations, e.g. for area calculations for cadastral administration, the error function of the Prussian cadastral instruction 2 should be used; not because the error regulations in this instruction are valid as a model. On the contrary. The error provisions in this instruction are outdated and require a thorough review. If the Prussian cadastral administration were to issue supplementary regulations again in the near future, an addition should also be made to this part of the instructions. Renewal takes place. Until then, however, anyone who carries out area calculations for the cadastral administration must comply with the above regulations; therefore the same may be used for the purposes of this treatise. The error formula for the deviation between two area calculations based on the same measurement can be used for this purpose. Strictly speaking, this is not entirely permissible, since the mapping errors must be taken into account when calculating from original numbers and the subsequent graphical calculation. The requirements of this formula are as a result a bit too high for our purposes. The error function is:

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where F and a are expressed in Ar.

For a 10 cm square, on a scale of 1:1000, there are one hundred ares. The permissible deviation according to the above formula is ±0.79%, while with the planimeter a deviation of ±0.82% was obtained. If one now takes into account that the deviations in the error formula would have to increase for the reasons mentioned above, it can be seen that the errors of the planimeter remain within the permissible limits, that the instrument is suitable for the purposes of cadastral administration and thus for all other purposes which a planimeter comes into question can be used without any problems.

The instrument also has a number of other advantages, including its handiness and stability, as well as the amount of work it can achieve. It is therefore desirable that this planimeter be widely used.

1] Zeitschr .f . Instrumentenkunde 1911, p.65 f.f and p.173 f.f

2] A figure is included in: Jordan, Handbuch des Vermessungswesens (Handbook of Surveying) Vol II, 8th edition, p. 148

3] Inventor's instructions