by J. Schnöckel in Berlin.

The instrument [1] described below is primarily intended to quickly and accurately evaluate the curves drawn by recording devices. It avoids the need for averaging or other calculations, because when "optically tracing" a curve, a thermogram, barogram, etc., the planimeter guided over the recording strip indicates the mean height (ordinate) directly, without the need to also trace along the Ordinate and abscissa axes (zero line).

As is well known, recording strips can be evaluated with all precision planimeters by tracing around the curve together with the axes as a surface, calculating the content and dividing it by the abscissa, most appropriately with the roller planimeter, which is expensive and whose tracing arm must be set to the abscissa length each time to avoid calculation as much as possible. However, as the optical planimeter can be used to easily measure strongly wave-shaped curves up to 30 or 40 cm long in one or two minutes with an average error of less than 0.1 mm, its performance exceeds that of expensive devices. It also has other more or less valuable properties, such as: various types of area calculations, the automatic drawing of the mean integral curve, hyperbolas, etc. and the possibility of perspective measurement of large areas (leather measuring machine). However, since other, sometimes more suitable instruments are already commercially available for these purposes, less emphasis has been placed on this in the following description.

The 20 to 25 cm long apparatus [2] shown in two different versions in Fig. l consists,
similar to Prytz's rod planimeter, of a rod or frame *aa* with thread *f*,
index *i*, the sharp-edged wheel (or cutting edge) *r* and a partially perforated, double
plane mirror *s* attached above or next to the wheel on a bridge, which is vertical when
in use, but can be folded down onto the frame for packing. To solve drawing tasks, the
bridge has a draw spring. The handling differs from other planimeters because there is no
tracing pin. This is replaced by the "optical tracing point"
that still needs to be defined in more detail.

For use, the instrument is placed on a paper base *p* (Fig. 2 and 3), which is
designed as follows [3]. You measure the distance of the wheel axis
from the index *i* (equal to *Ri*) and mark the point *R* at this distance
from the edge of the paper with a small circle, draw the line *l* which can have any
curves at the ends, parallel to the edge at twice the distance and draw a perpendicular
from *R* on *l*, which is extended. A less important bifurcated curve
intersecting the extension is designed based on a simple formula given in the second
section.

This system is placed on the registration strip (Fig. 2) in such a way that the edge
lays along the ordinate of the starting point *U* of the curve *k* to be evaluated
and the latter falls into the extension of the perpendicular to *l*. The instrument
must now be placed on the system with the wheel in *R* so that the thread is in the
extension of the plumb line and the index *i* on the planimeter falls on *U*. If
you look from any point *B* into the broken plane mirror *t*, the image of the
intersection of *f* and *l* appears in *U*, which may be called the "optical
tracing point". At the beginning of the process you use the lower part of the mirror and
only later the upper part in order to clearly see the index and the curve through the
fine interruptions in the silver coating. Even when viewing obliquely, the observer
notices two images of the intersection of *f* and *l*, which belong to a moving
and a fixed ray. The optical tracing point is the image that is further away from the
observer and instrument and is associated with the fixed beam, which is also
characterized by its clarity in front of the moving image. During navigation, the
apparatus and recording strips must be well lit and the thread must not be shaded by the
mirror. If the mirror is vertical, the optical tracing point does not change its position
when you alternately look at the top and bottom of the mirror surface. However, every
movement of the planimeter caused by the guide *c* (see also Fig. 1) changes the
position of this point, and it is

It is very easy to guide the instrument in such a way that it tracks along the curve
*k* from *U* to the end point *U _{1}* with the same certainty as
the hand might guide the tracing pin of a polar planimeter. The eye is on average
10 to 15 cm away from the tracing point, so it can see it within a good visual distance
and at the same time regulates the movement of the planimeter itself without any
difficulty. In the end position

The problem can now be considered solved, because the ordinate of the index, which is
now in *i _{1}*, is the desired mean ordinate (height) of the curve

If the system *p* is unfavorable, the thread *f* in its final position *RR _{1}*
(Fig. 2 and 4) deviates greatly (more than 10°) from its initial position

The planimeter only delivers theoretically correct results if the pen of the recording device moves, not on a circle, but on a straight line parallel to the ordinate axis. However, the resulting inaccuracy is not noticeable on the recording strips of meteorological devices of this type, since the arm carrying the pin is very long in relation to the wave height of the curves. In the case of strongly curved vapor voltage curves, this may need to be taken into account by applying a correction [4]. If you want to check the result by tracing from a different starting position, it is best to trace the curve in the opposite direction and average the results (see the last section).

In an adjusted optical planimeter, the axis of the wheel, which is perpendicular to
the thread line, passes through the extended mirror plane, which is perpendicular to
the base. If, under this assumption, *FRU* in Fig. 4 denotes the initial position
of the planimeter on the base *p*, then *U* is the optical image of *F* and
*FU = 2FR = x _{0} = 2 ξ_{0}*.

If the optical tracing point *U* passes to *U _{1}* along the curve

Then, because *F _{1} U_{1}* touches the path

or

from which follows through integration

After rotation around *R _{1}*,

As the parallelograms *FUHE* and *F _{1}MHE* have the same area and
Eq. 2) with Eq. 3) are identical, it follows that

If *L* is the center of *MI _{1}*, then the perpendicular must be

from which, by subtraction, *R _{1}L = RM* [5]. If

Since *RU* is constant equal to 24mm, the displacement *i _{1}L* depends
only on the inclination of the thread

Instead of making a shift in the direction *F _{2}R*, you can also influence
the same on the middle ordinate

which is calculated in column 3, can be read off on a scale parallel to *l*.

From Eq. 2) follow the equations for the curve q (Fig. 4), the mean integral curve
of *k* [6], which describes the wheel during optical travel,
with the parameter *x*,

It is often desirable to draw this curve, which graphically represents the growth of
the mean. This is done either with carbon paper, which is placed under the wheel, or,
as can be seen from Fig. 1, by adding ink to the wheel in a drawing pen or sleeve,
which consists of black, unwashable drawing ink, gum arabic, glycerine and water mixed
in proportions 0.35: 0.25: 0.3: 0.1. The planimeter then automatically draws
a deep black, evenly thick line on the paper, which dries quickly and lasts well.
For example, if *k* is a straight line, then *q* becomes a hyperbola with the
asymptotes *l* and *k*, which follows from Eq. 5).

The planimeter is therefore particularly suitable for precisely drawing hyperbolic
and other isopleth tables, which are widely used in technology. If you want to draw a
hyperbola, of which a point *P* and the lines *l* and *k* are given as
asymptotes, you bring the locked wheel to *P* and turn the planimeter until the
optical tracing point *k* hits. After releasing the escapement screw, *k* is
driven optically while the wheel writes the hyperbola automatically.

Eq. 3a) shows that the apparatus can also be used as a linear planimeter, because
the area below *k* (Fig. 4) is the product *HI.LL _{1}*. The figures
formed only by curves can also be calculated in a similar way. If one imagines the area
from a point

The simplest, although not quite as accurate, method for calculating curvilinearly bounded areas follows from Eq. 1):

Because *U* is identical to *U _{1}* (Fig. 5) it follows

As the mirror can be used from both sides, the planimeter can also be turned over
so that the frame falls over the figure *k* (Fig. 5). Using a slider with a mark,
which is held in two grooves within the frame, you set *RU = ½FU*
approximately equal to 10, choose the edge of a ruler at a distance of *RF* equal
to 10 as a replacement for the line *l* and go around *k* optically, so that
the image of the thread intersects the edge of the trace in *F _{1}*.
The area is ten times the difference in reading on the scale at

With the rotor as a mark, the apparatus can be used effectively like a Prytz rod planimeter. Here, the wheel marks in the paper can be avoided if the mirror image of the thread is cut with a prismatic ruler.

An approximate perspective-optical area calculation of larger areas of several
square meters with the small instrument can easily be carried out in the following
way. Cover the paler side of the mirror facing away from the observer with paper
except for a small, three millimeter wide opening *S*, place white cardboard on
the edge of the table (Fig. 6) and draw the line *l* through *F* parallel to
the edge. A larger piece of surface lying on the floor with the center in *U*
is optically traced with the planimeter *FRS* standing on the table level using
the small mirror opening *S*, after *U* is connected to the circumference of
the figure by an arbitrary line. As in Fig. 5, let
*F _{1}R_{1}U_{1}* be the end position.

If you set *n = FR/NU*, then this is the area of the piece lying on the ground:

The points *FRNU* are determined once and for all and a round number is chosen
for *FR/n ^{2}* so that areas can be read from

The correctness of the latter type of calculation can be proven using Figures 4 and 6. In Fig. 6 behaves

This ratio of x:ξ exists generally, so that for Eq. 1) is to be written

If one replaces *x* by *ξ* on the left, then partial integration follows

If the curve closes, then

and because the image of *l* at the initial point passed through the center
of the figure *k*, approximately

As *RR _{1}* also roughly halves the remaining figure

A contraction results in Eq. 6).

A number of screws allow the thread, wheels and mirror to be adjusted. You can check
whether the mirror is perpendicular to the base according to what was said in the first
section or by ensuring that two points that are the same distance (15 cm) from the wheel,
e.g. *F _{1}* and

When measuring recording strips, it is actually unnecessary to adjust the wheel
because the error, if the index *i* falls within the extension of the thread, is
proportional to the length of the abscissa and can therefore be correctively added to
the result. If the wheel and thread form small angles with the mirror normals, and as
a result the index *i* deviates to the right (left) by *Δ _{0}i*
(e.g. 0.90 mm) after optical travel for the abscissa

One determines *Δ _{0}i* empirically as half the difference between the
results of two optical measurements of a wave-shaped curve

In contrast, precise adjustment is required to draw hyperbolas. After the wheel has been brought into the thread line, the axle is placed perpendicular to it and whether this has been achieved is checked by moving a thread point along a straight line without the wheel leaving it. Now place the mirror approximately vertically and place the bridge on the lower part so that the mirror plane points to the wheel axis, making sure at the same time that the mirror image of the thread falls into its extension. The mirror holder hits a screw that has to be adjusted so that the mirror is immediately in the correct position when opened. The Planimeter draws hyperbolas almost as precisely as a compass draws circles.

The instrument gives good results as a linear planimeter. The average error is only ±0,3% for medium-sized areas (1 qdm), and little more for small ones, while the calculation according to Fig. 5 is less precise and particularly dependent on the nature of the card paper. On a flat surface, the average error for small areas is ± 0,6%, but increases slightly with the size of the area.

The perspective calculations according to Fig. 6 suffer from a strong parallax
of the optical tracing point and from estimation errors regarding the correct position
of *U*, so that errors of 1 to 3% are to be expected. For this reason, the
determination of the moments of small figures, which can be achieved by suitable
integration of Eq. 7) is made possible, and enlarging and reducing figures is not
mentioned.

Berlin, Agricultural University, January 1911.

*The instrument, designated the "optical universal planimeter", has been submitted for a patent and, as it is not yet available in stores, can only be obtained from the author, Berlin N., Invalidenstraße 42, Königliche Landwirtschaftliche Hochschule. The price including the case is 48 M.**Instruments of the geodetic department of the Royal Agricultural College in Berlin.**Such a system (Fig. 3), produced photographically, is included with the planimeter.**A larger apparatus for evaluating such diagrams, in which the strip under the instrument is moved with the help of a crank, is patented by Siemens & Halske A.-G., Berlin (D. R. P. No. 214 195 and 214669).**Fig. 4 is drawn partially distorted for clarity.**In a very simple way it also becomes the integral curve**η = ∫y.dx*found point by point.