The instrument shown in Fig. l, called a compensation rod planimeter [1], is for the precise calculation of the area of curvilinear figures and for the evaluation of registered curves. It resembles in shape the Prytz rod planimeter [2], which has been known since 1886; an invention that has often been discussed in specialist journals. This communication is intended to give a description and theory of this new planimeter, and also to consider its advantages and disadvantages in comparison with other systems, paying particular attention to their accuracy.

This rod planimeter differs significantly from the Prytz instrument in that for it
to move around a figure with the tracing pin, the rod *a* is placed over the edge
of a roller *r* (Fig. l upper and lower) or on a ball (middle instrument). As it
rolls, the distance between the rollers and the tracing pin does not remain constant
(as with Prytz), but changes continuously. At the end of the rod is a reading device
with an index line *e* and the rod rests on the paper at three points: on the
tracing pin, the wing handle and the edge of the roller. The roller is cylindrical,
with a protruding sharp edge which is held in a groove *s* in the rod, while the
cylinder presses it laterally against the groove, so that it only rolls in the
direction of the rod and the planimeter cannot fall over.

[*The scale 'eb' shown in Fig. 1 represents an earlier version. Also, as can be
seen from the description below, the fixing needle inserted into the paper at b is
now gone.*]

For area calculation, e.g. plot 8 (Fig. 1), place the tracing pin on any point on
the contour so that the rod does not cover the area and is directed approximately
towards its center, while the right edge of the roller rests at one of the marks on the
groove *s*. Align the rule [3] *eb* with the scale
*c* against the instrument so that the zero division is approximately
perpendicular to the index line *i* of the rod. The scale rule has a needle point
at *b* (similar to the pole of a polar planimeter)) which rests on the paper, and
you can rotate the scale around *b* to push it to the side while tracing the
surface. After a clockwise circuit, bring the scale back to the index line *i* of
the rod by turning at *b*, and take a new reading. Ten times the reading is the
simple or double the area of the figure measured, depending on whether the roller was
set to the left or right mark on the rod.

However, at least for larger areas and precise calculations, this value still
requires a small correction, which depends mainly on Φ itself, but also on a
ratio *v*. *v* is estimated from the position of the rod in relation to
the figure and is the ratio of the diameter of the area in the direction of the rod to the diameter
in the direction perpendicular to the rod. *v* is equal to one for all
approximately round or square figures. For the multiplication
constant 10 (roller distance from the pen 20 cm) the following correction table
applies, which is only partially reproduced here. The arguments are the area Φ
and the *v* to be estimated in 1:1000, the corrections mean square meters.

The ratios *v* are listed in column 1 from 4 to 1/3. It is well known that with
the polar planimeter there are certain favorable configurations that can be expected
to produce particularly precise results, and that is also the case here. Table 1 provides information
about this important question and makes it clear that with larger, highly irregular
or very elongated figures one can expect a more precise result if the rod is placed
in the direction of the largest diameter of the surface, i.e. if *v>1*, because
an error in estimating *v* then only means a very small error in the correction.
As an example of the calculation process and the use of Table 1, a calculation of
plot 8 in Fig. l is used, the area of which on the scale l:1000 has the target value
of 4347 sqm if *v* was estimated at 1/2:

Ten times reading: 4325

Correction: 10

Area: 4335

A significant error in *v* would change the result by at most 5 to 8 square
meters or ± 0.2 percent. The roller regularly returns to the mark for small
figures, but occasionally not exactly to the mark for larger figures. This deviation
Δ is estimated in mm in order to use it and the area Φ to derive a
correction from the following table:
*[For the 24 calculations on the map in Fig. 1, Table 2 was used only once.]*

Here the value *A* in millimeters is regarded as negative if the roller is
closer to the tracing pin after a circuit and vice versa. Strictly speaking, even with
a large *P*, it becomes zero when the bisecting line of the angle that the initial
position makes with the final position of the rod passes through the center of gravity
of the figure. Therefore, when calculating large areas (larger than the size of your
hand), you can first make a superficial test tracing to see whether *Δ*
almost disappears. Such an experiment would correspond to a similar test to be carried
out with the polar planimeter to see whether the area can be calculated. On the
other hand, a look at Table 2 shows that this correction for negative *d* is small
(only about 0.1 percent of the area) and can be neglected. This is based on the simple
rule of placing the rod in such a way that its extension either halves a larger figure,
or that the part of the area facing the computer becomes the larger, up to 2/3 of the
entire area.

An advantage over the polar planimeter is the quick change of the multiplication
constant 5 or 10 without having to loosen screws. Figures up to 30 cm long can be measured.
If they are larger than about 2 sq. cm, or so irregular that they cannot
be roughly estimated, they must be measured separately in two parts. If a second
compensating tracing is to be made, choose a position of the rod that is approximately
perpendicular to the first in order to eliminate any inaccuracies resulting from an
incorrect estimate of *v*. With the polar planimeter, a repeat
measurement is often made in order to make the reading errors as harmless as
possible by moving around small figures twice in a row with a fixed pole position, but
this achieves little advantage because the roller makes the same path each time. This
latter is not the case with the rod planimeter, so the advantage is fully exploited.

For calculations on poor paper, or if there is no space for the roller on the map
because the figure is on the edge, run the roller on a 5 x 10 cm zinc plate included
with the planimeter. It is also sufficient for large areas up to 20 cm long because the
roller only covers half the travel of the tracing pin compared to the polar planimeter.
Imperfections which have an unfavorable influence on the guidance of the tracing pin
cannot occur with the planimeter rod, as they occur with various planimeters, but
especially with the Lippincott planimeter from the inclined position from the tracing
arm to the pole arm. With a tapered map scale, e.g. 1:2000, the reading difference
*d* must be on the prismatic scale instead of 5 or 10 with 20 respectively be
multiplied by 40. But if the constants are no longer round numbers, you choose other
marks on the rod or a suitable measuring scale.

Experiments show that strong bending of the rod has no effect on performance and accuracy, so that it can be safely handed over to untrained people.

The precision investigations that follow are based on calculations of the map shown
in Fig. l which has 24 different sized, very irregular figures and an area of almost
12 hectares at 1:1000. To calculate the target values, the average (column 2 of the
following table) was formed from three circuits with the free-floating precision
spherical planimeter Coradi No. 612 (price 150 M.), a collection item in the Geodetic
Department of the Agricultural University, set to the constant l. Each figure was then
traced once (very small figures twice) with the Coradi compensation planimeter No. 2002
(price 65 M.), whose roller skew has been eliminated. The summation of the results
resulted in a deviation of -225 per 119,605 square meters, which was distributed
proportionally. The difference between the reduced values and the target in column 2 is
shown in the third column of the table. In order to be able to compare the accuracy of
the rod planimeter with that of the chosen compensating planimeter (Fig. 2, below), each
figure was measured just once with rod no. 13, with the roller set to the
multiplication constant 10, or for very small figures 5, and the deviation
from the target (column 2) entered in column 4 each time. Two further calculations carried out
under the same conditions with bars no. 13 and 14 revealed the errors in the last two
columns. Plot 20 was always divided into two parts due to its shape. The area summation
showed errors of -63 respectively for the first two calculations. 6 square meters
against 119605 square meters. In the third calculation made a few weeks later, the card
had fallen by 0.1 percent, while the final deviation was -127 square meters. Here the
rod was intentionally placed unfavorably, i.e. with *v* < l.

Each of the calculations in columns 3 to 6 took almost two hours, so each result took about 4 minutes. If μ denotes the average error of a single area calculation in square meters on a scale of 1:1000 and Φ denotes the area in square meters, the above compilation results in the following formula both for Coradi's compensation planimeter (column 3) and the new rod planimeter (columns 4 to 6).

The formula generally expresses that both instruments can be regarded as equivalent in terms of their accuracy. The accuracy of the expensive precision planimeters, which cost around 150 M., is considerably greater, of the order

The results of the above calculations are shown graphically in Fig 2. The area Φ in square cm is entered as the abscissa and its calculated mean percentage error as the ordinate. Curve 1 parallel to the axis represents approximately the error of a Prytz rod planimeter after a circuit made from the estimated center of gravity, curve 2 represents the average error of the Coradi No. 2002 compensating planimeter in percent of the area, curve 3 (the solid line) that of the new rod planimeter and curve 4 that of the free-floating precision spherical planimeter of Coradi, as it resulted from the calculation in column 2 for a single tracing.

**Theory of the compensation rod planimeter and the determination of the mean
ordinate of registered curves.**

The way it was created, the compensation rod planimeter can be viewed as a
simplification of the "optical planimeter" [4]. In Fig. 3, let
*R* be the roller and *U* the tracing pin of the rod at the starting point
of the curve *UU _{1}* to be evaluated. If you imagine the reflective plane
of the optical planimeter above the roller perpendicular to the rod and to the paper,
then

With the rod planimeter, the rod touches the roller (ball) in R, so that the rod
moves in its own direction as quickly as the roller and the roller must always be in
the middle between the fixed point *F* and the tracing pin. When the rod rotates,
the geometric location of *F* in the differential sense is a piece of a circle,
whereas with the optical planimeter this curve can be chosen arbitrarily.

However, the formulas developed in [4] cannot be easily
applied to the rod planimeter because, firstly, the line *l* was assumed to be
a straight line and secondly, at least when using a sphere, *RF* is slightly
larger than *RU*. In this case, the rod touches the ball, which sinks slightly
into the groove, at two points, so that a general statement can be made.

The actually circle-like curve *l* from the equation *z = f(t)* is
arbitrarily distorted in Fig. 3 in order to give a clear, geometric explanation of
the following theory. As the involute of the roller path *q*, whose
coordinates are *N* and *F*, it has return points, the number of which
depends on the shape of the curve to be evaluated *y = f(x)*. According to Fig. 3

If you replace *dξ* on the left; according to Eq. 1), then

The figure now results in *η-z = n(y-η)*. If one inserts *y* or *ξ* from
this equation and from Eq. 1) into the last two terms of
Eq. 2) one, it follows

or by decomposition

If you replace *dx - dξ* according to Equation 1) and integrate, you
will find the area formula

This formula can be explained geometrically as follows for *n=1*
(sharp-edged roller):

This proves that

If one takes into account that the line *l* can in reality be viewed
almost as a circle with a fixed radius, one only has to connect the points
*R* and *R _{1}* marked by pressing
on the rod [5], this line around

If, on the other hand, *UU _{1}* is a closed figure with the
content Φ (Fig. 4), i.e.

and Eq. 4) becomes

or Φ = sector *UFF _{1}*. If

Table 1 was calculated by graphically integrating differently shaped figures and
with changing *v* with the aid of interpolation, so that it also includes the
system of the scale *eb* (1: m.1000) shown in Fig. 1 and the reduction of the
straight line *ii _{1}*, which the circular sector

A theoretical series development [6] for the correction table 1 has been omitted
at this point. If the ball sinks halfway into the bar frame, then n = ∞ and
the rod planimeter becomes a Prytzian bar planimeter, whose performance greatly affects
the remaining figure *q*. In the case of the ball bar shown in Fig. 1,
however, n = 1.05 and the influence of *q* is harmless.

Although the accuracy of the most expensive precision planimeters costing over 100 M. cannot be achieved with this new planimeter, it has been shown that the rod planimeter produces results that are just as accurate as the compensating polar planimeter that is available on the market but is twice as expensive.

Berlin, Royal Agricultural University, March 1911.

*The planimeter has been registered for a patent, but is not yet commercially available and is currently only available from the author, Berlin N, Invalidenstraße 42, Geodätische Abteilung der Kgl. Landwirtschaftlichen Hochschule. Price of the rod planimeter with roller (Fig. 1) in a case 28 M.**See the treatise "The Prytzsche Stangenplanimeter" by E. Hammer, this periodical 15. P. 90. 1895.**Each planimeter is accompanied by the scale shown in Fig. 1, covered with white celluloid, which is provided with a tongue e serving as a gauge and a metal attachment, etc.**Cf. the author's treatise "An optical planimeter". Zeitschrift für Instrumentenkunde, June 1911, Vol. 31, p. 173. Also A. Schmidt, "A planimeter for determining the mean ordinates of arbitrary sections of registered curves". This magazine 25. P. 261. 1905.**The sharp edge of the roller presses a little into the paper each time.**A similar development is probably possible here as C. Runge wrote in the journal. f, surveyor 24. p. 321 et seq. 1895 for the Prytzsche Stangenplanimeter.*