1. Introduction

I recently had the good fortune to acquire a Schnöckel Compensation Planimeter. It is complete, although in a rather battered case.

At first glance, this looks like a hatchet planimeter. But it has no hatchet and operates differently. Other planimeters: all polar planimeters, linear planimeters and hatchet planimeters, operate with a fixed length tracing arm. The effective length of the tracing arm of this planimeter varies continuously while an area is being measured, because the arm is rolling on the ball.

I can't find much about Schnöckel on the Web, but Andries de Man has a terrific collection of documents about slide rules, calculators and planimeters in his web page of "Original Documents on the History of Calculators", including a pamphlet about Schnöckel's Kompensations Kugelplanimeter Type 2. I translated the pamphlet to see what it says, and my rough translation is in the Appendix.

Other than that,

• Schnöckel applied for German patent 238500 on 25 November 1910. (My translation into English is here).
• Schnöckel published a paper describing his instrument in Zeitschrift für Instrumentenkunde, June 1911, Vol. 31, p. 173-179. (My rough translation into English is here)
• Wolff published a critique of the planimeter "Das Schnöckelsche Kompensationsplanimeter" in Zeitschr. f. Vermessungswesen 1914 (XLIII), S. 321. (My translation into English is here).
• Galle mentions it in his book Mathematische Instrumente published in 1912.
• Willers, Friedrich Adolf: Mathematische Instrumente, Sammlung Göschen, Band 922, Berlin und Leipzig: de Gruyter 1926 briefly mentions the optical planimeter.

And Detlef Zerfowski has kindly provided references to three earlier papers:

• Schnöckel, J.: Apparat zur Bestimmung des Flächeninhalts, des statischen Moments, Trägheitsmoments und beliebiger anderer Momente krumlinig begrenzter ebener Figuren, Zeitschrift für Math. und Phys., Vol. 49 1903, pages 372--
• Schnöckel, Johannes: Graphische Integration, Zeitschrift für Vermessungswesen, Vol. 32 1903, pages 129-142. Available here
• Schnöckel, J.: Die Steigerung der Genauigkeit graphischer Berechnungen mit Hilfe von Parabeltafeln, Zeitschrift für Vermessungswesen, Vol. 34, no. 18 June 1905, pages 414-417. Available here

I have been unable to locate any biographical information about Schnöckel. He is not amongst the 730,000 persons in the Deutsche Biographie

I have borrowed Andries' Javascript to identify the parts of my planimeter, as follows:

• Ball (Kugel);
• Ball-stop with index mark (Kugelanschlag mit Indexstrich);
• Tracing arm, resting on the ball and the cursor, with a groove in which the ball can roll freely (Fahrstab);
• Starting mark (Anfangsmark der Kugel);
• Transparant cursor (Fahrplatte);
• Cursor handle (Handgriff der Fahrplatte);
• Measuring scale, (note that there is no physical connection between the tracing arm and the scale)(Ablesemaßstab);
• Pin around which the measuring scale rotates (Spitze des Ablesemaßstabes);
• Strut of measuring scale (Stütze des Ablesemaßstabes);

Wolff provides a schematic of the planimeter:

The following operating instructions are mentioned in Schnöckel's and Wolff's papers:

• Before you start calculating the area, check that the work surface is level by rolling the ball on it. If it deviates from a straight line, the surface is crooked and this needs to be remedied.
• Place the rod a on the ball k, which rolls in a groove underneath the rod.
• Put the tracing mark f on the border of the area to be measured, over a corner point or at the end of a short line l, in such a way that the area is not covered by the planimeter. Ensure the planimeter points more or less towards the center of the area, and in the direction of the largest diameter for elongated figures.
• Position the ball below the special mark on the rod.
• Put the measuring scale against the ball-stop, and align the scale with index line i, preferably at 0 or 1.
• Press the pin n at the corner of the ruler lightly into the paper and carefully turn the ruler to the left, so that it won't interfere with the rod while tracing the border.
• Trace the border clockwise.
• When the tracing mark has been returned to the starting point, rotate the scale back to the index mark and take a second reading.
• Multiply the difference between the two readings by the corresponding constant to obtain the area.

And, as Andries adds:

• Apply a correction which has to do with the shape of the area, expressed as the ratio between the length of the area in the direction of the rod and the width of the area in the direction perpendicular to the rod. A table is provided for some rational values of this ratio.
• Apply a correction which has to do with the distance between the final position of the ball and its starting mark on the rod. This distance is usually close to zero. A second table is given for this correction, with the distance varying between -0.75 and 0.75 mm. So what are we talking about? ..

These planimeters were made commercially but seem to be rare. I can locate only two others:

1. In the Department of Mathematics and Physics, University of Stuttgart. Item MI_8: "Kompensations-Kugelplanimeter nach Schnöckel"

2. In the, unfortunately small, photo in Andries de Man's web page of "Original Documents on the History of Calculators".

This is an interesting pamphlet. Andries put the German text up on his web page. My rough translation is in the Appendix.. It mentions:

No. 1. Compensation planimeter with ball bearing, with a fixed, approximately 30 cm long rod, graduation of the reading scale plated in white celluloid, constant table in a case, with 2 precision steel balls, a magnifying glass and instructions. Price with free shipping Marks 25.-.

No. 2. Compensation planimeter with ball bearing, with adjustable rod, adjustments for 7 different rod settings that correspond to convenient multiplication constants in common map scales; table of constants in a case, otherwise as with No. 1. Price with free shipping Marks 30.-.

We can see from this that my planimeter is the Type 1 model and the two others are the Type 2 model.

It goes on to explain the Type 2 model and also mentions that "The ball, which has a certain degree of roughness, should not be dragged over the paper, but should always be allowed to roll over it." This seems to me to point to a potential weakness. The sharp blade of a hatchet planimeter offers considerable resistance to sideways slippage. The only thing preventing slippage with this planimeter is the friction between the ball and two contact points with the fairly light rod. Using it requires a delicate touch.

My planimeter has the serial number 183. It looks as though there may be a serial number on the rod in Andries' picture: possibly 4XX. And the brochure advises that "Over 50 planimeters with ball bearings were delivered to the Royal Prussian General Commissions". But I don't believe that more than a few hundred were made.

There may also be a reason why you see so few of them today. The plastic with which the tracing plates were made has not aged well. Mine has become discoloured and brittle. Rust from the screws has spread through the cracks in a couple of places.

The Stuttgart cursor (below) looks even more crazed than mine. Broken planimeters would very likely be discarded.

Some of the details on my tracing plate are mysterious. The tracing point is marked as the intersection of two lines (x marks the spot) but why is one of the lines that strange shape? Wolff labels it as g in his schematic but doesn't offer an explanation. There is also a little scale engraved on the end of the plate, which is about 5mm divided into tenths. Woolf notes that "To increase the reading accuracy, the inventor 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".

Andries' pamphlet mentions: The new "Instructions for the practical use of compensation planimeters with ball bearings", knowledge of which is assumed here .... If only we could find a copy.

Schnöckel described his instrument at length in Zeitschrift für Instrumentenkunde, June 1911, Vol. 31, p. 173-179. (My rough translation is here ). The paper includes this illustration of three models, none of which matches the surviving examples.

He mentions that "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 in a case 28 M." So planimeters with rollers were made, but I cannot find an example anywhere on the Web.

He also advises that "The way it was created, the compensation rod planimeter can be viewed as a simplification of the "optical planimeter". "However, the formulas developed for it cannot be easily applied to the rod planimeter ..." I also checked this article and, for what it's worth, my translation is here.

Schnöckel invented two other types of planimeter: the Optical Planimeter and the Vector Planimeter.

The Vector Planimeter was patented in 1925 and is described here.

The "Vektorplanimeter" ( J. Schnöckel, "Das Vektor-Präzisionsscheibenplanimeter", Z. f. Instrumentenkunde 46 (1926) S.67-70 patent

He patented his "Optisches Planimeter" (Optical Planimeter) in 1910 and described it in Zeitschrift-für-Instrumentenkunde Vol 31, March 1911, 65-71. A rough translation is here This illustration is from his paper:

I can find only one example on the Web, in the Department of Mathematics and Physics, University of Stuttgart, Item MI_3: "Optisches Planimeter". It appears to be missing its mirror.

Galle p129 provided the following summary of the optical planimeter in his book, and also referred briefly to Schnöckel's Compensation Planimeter:

"J. Schnöckel's optical planimeter differs from others mainly in that it does not have a tracing pin. Rather, above a sharp-edged wheel, which is attached to the end of an elongated rectangular frame, there is a vertical mirror, the plane of which contains the axis of the wheel and the point of contact of the wheel. The instrument is mainly used to measure recording strips and determine the mean ordinate. For this purpose, an index located at the front is placed at the starting point of the curve, and a straight line is drawn backwards on a piece of paper underneath at the same distance from the wheel support point, parallel to the ordinates of the curve.
The instrument itself is set up in its longitudinal direction parallel to the abscissa axis, so that a thread stretched in this direction intersects the straight line l drawn on the paper at a point F that is symmetrical to the starting point with respect to the wheel. This point is viewed through the partially transparent mirror and therefore coincides with the mirror image of the starting point of the curve when the instrument is in the initial position.
The instrument is now moved in such a way that the intersection of the thread with l always coincides with the curve points that appear in the mirror. When the end point of the curve is reached, the index point corresponds to the middle ordinate. After locking the wheel, the instrument is now rotated until its thread direction passes through the first support point of the wheel in the initial position.

"If x, y are the coordinates of the optically adjusted curve point, ξ, η of the contact point of the wheel, then if the straight line l corresponds to the value x = 0, then x = 2ξ. As the direction of the instrument (more strictly of the thread) affects the path of the roller at every moment, then

where τ is the angle of the tangent with the abscissa axis. From this it follows

"As y_m is the value of the center coordinate sought, then considering x = 2η, x₀ = 2η₀, ξ = y_m. The use of the instrument for recording the mean integral curve or, in special cases, for drawing hyperbolas, is explained in Zeitschr. für Instrumentenk. 31, 69, 1911.

"The same inventor created a compensation planimeter, which essentially consists of a simple rod, with a tracing pin, that rolls on a wheel. The sharp edge of the wheel rests in a groove on the rod, when the rod is placed on it.
To prevent the wheel from tipping over, it is extended laterally into a cylinder that rolls on the base. (In other designs, the cylinder is replaced by a pair of wheels or a ball.) Another support point for the rod is attached next to the driving pin at one end. The other end of the rod ends in an index against which a scale is placed approximately perpendicular to the rod, but which can be turned away by moving around a needle tip at one of its ends so as not to hinder the movements of the rod. After completing the trace, the scale is again placed against the rod index in order to measure the distance the index has moved (before and after the tracing), similar to the distance between the marks with the Prytz planimeter. The wheel is set at a mark on the rod and after the tracing returns exactly or at least almost to this initial position.

"In the latter case, a correction must be made. The theory (see Journal f. Instrumentenk. 31, 173) follows that of the previous instrument in that the rod moves in its direction twice as fast as the wheel. The accuracy has been found to be approximately equal to that of the Coradi compensation planimeter and significantly exceeds that of the Prytz instrument, while remaining somewhat below that of the Coradi free-floating precision spherical planimeter."

Willers (page 114) briefly mentions that: Schnöckel's optical planimeter can also be counted among the integraphs (Zeitschr. f. Instrumentenkunde 1911, p. 65), which records the curve

on the abscissa ξ = x/2. A curve that can perhaps be called a mean integral curve, since it continuously indicates the mean ordinate of the integrated area. Of course, you can easily construct the actual integral curve from this curve.

Appendix

No. 1. Compensation planimeter with ball bearings, D.R.G.M. and D.R.P. with a fixed, approximately 30 cm long driving rod, graduation of the reading scale plated in white celluloid, constant table in a case, with 2 precision steel balls, a magnifying glass and instructions. Price with free shipping Mark 25.-.	Planimeter No. 1 in operating position with case.
No. 2. Compensation planimeter with ball bearings, D.R.G.M. and D.R.P. with adjustable rod, adjustments for 7 different rod settings that correspond to convenient multiplication constants in common map scales; Table of constants in a case, otherwise as with No. 1. Price with free shipping Mark 30.-.	Adjustable travel rod of planimeter Nr. 2.

The following may be considered advantages:

1. The free choice of the multiplication constants with the most convenient adjustment of the rotor on the driving rod. (Compare figure above.)
2. The possibility of taking into account the paper thickness of a card (or its extent) by allowing a runner displacement of 0,2 mm to occur for every 0.1% surface thickness.
3. Increasing accuracy by shortening the travel arm, especially in small areas.

"The instructions for the practical use of compensation planimeters with ball bearings", knowledge of which is assumed here, also applies in its entirety to model No. 2. This differs from No. 1 in that the ball stop cannot be detached from the rod, but can be turned firmly by hand as a runner at any point. It encloses the rod as a U-shaped piece of metal, which has a white celluloid plate at the bottom, the ball stop with an index mark i for reading the area measurement. The illustration shows the stop with the index adjusted to one of the graduation marks on the travel rod as required. In the table of constants below, reference is made to these lines in column 3.

If you had placed the runner on line number 3 on a map scale of 1:n, e.g. 1:1500, and read off, after circling, an area 2136 (estimated at 6), the area would be k=40 x 2136, i.e. 85440 square meters. To mark additional lines on the moving rod, determine the distance a of the line you are looking for from line number 1 using the formula:

but choose k so that a is at most 10 cm long. Before you start calculating the area, check the table top is horizontal top by rolling the ball on it. If it deviates from a straight line, the table is crooked, which needs to be remedied.

In all rod settings, the simple rule applies (instructions on page 4), which eliminates the need for a test drive, that after setting up the rod, the runner does not fall into the interior of the figure and in the direction perpendicular to the rod it is no more than a hand span (approx. 18 to 20 cm ) wide. If you don't take this into account, firstly the result will often no longer be readable after the tracing (over approx. 5 sqdcm), but also, according to the planimeter theory, an exact result can no longer be guaranteed at all. When tracing surfaces that are unfavorably shaped planimetrically, the so-called roll skew, which cannot be completely eliminated as with the polar planimeter, can be noticeable in the result - even if only slightly. If the accuracy requirements are high, this error (instructions on page 6) is eliminated by a compensating tracing with a second rod position that is approximately perpendicular to the first and also approximately halves the area, and the average is formed from both results. During the tracings, the handle of the travel plate can be easily rotated.

It should also be mentioned that when calculating the area of distorted leveling profiles with the length scale 1:n, e.g. 1:1000 and the height scale 1:m, e.g. 1:200 in the above formula for a, you only have to replace the size n² with n*m, so that, for example, for line number 1 (i.e. a=0), k = n*m/50000.

The ball, which has a certain degree of roughness, should not be dragged over the paper, but should always be allowed to roll over it. Any rusting of the same is harmless. If the tip of the reading scale breaks, loosen its retaining screw, drive the pin out by hitting the break point with a hammer and replace it from the "stock".

Over 50 planimeters with ball bearings were delivered to the Royal Prussian General Commissions.