Reference / Paper · 2023
A Contribution to the History of Planimeters (Vossepoel & Zerfowski, 2023)
Slide Rule Gazette 23 (2023): more than 30 lesser-known planimeters of 1814–1884.
Reference / Paper · 2023
Slide Rule Gazette 23 (2023): more than 30 lesser-known planimeters of 1814–1884.
Slide Rule GAZETTE Issue 23 Autumn 2023 A Contribution to the History of Planimeters – lesser-known instruments and variants Detlef Zerfowski & John Vossepoel Table of Contents 1. The challenge to structure the topic .................................................................................................................... 1 2. Early usage of the term “Planimetry” ................................................................................................................. 2 3. Bryon’s Planimetre Graphique, France, 1814 and 1826 ..................................................................................... 2 4. Gelinski’s Calculateur, France, 1815 .................................................................................................................. 4 5. Zaręba’s (Zaremba) Planimeter, Poland, 1817 ................................................................................................... 5 6. Laur’s Olarithme, France, 1819 .......................................................................................................................... 7 7. Colberg’s Planimeter, Poland, 1820 ................................................................................................................... 8 8. Wießner’s mathematical instrument, Jena Germany, 1823 & 1828 ................................................................. 10 9. Tenner’s Planimeter, Germany, 1828 ............................................................................................................... 12 10. Littmark’s Pollettcirkel, Sweden, 1831 & 1857 ............................................................................................. 13 11. Stabrowsky’s Planimeter, Prussia, 1835 ......................................................................................................... 14 12. Lalanne’s Arithmoplanimètre, France, 1839 .................................................................................................. 14 13. The Computing Scale, England, 1839 ............................................................................................................ 15 14. John Miller’s Pantoscale, 1841 ....................................................................................................................... 16 15. Roulette de Dupuit, France, 1844 & 1878 ...................................................................................................... 17 16. Hennon’s Polygonometre, France, 1846 ......................................................................................................... 18 17. Godfray’s Agrometers, England, 1848 ........................................................................................................... 18 18. Källström’s Planimeter, Sweden, 1852 ........................................................................................................... 19 19. Gierer’s Planimeter, Germany, 1854 .............................................................................................................. 20 20. Burman’s Planimeter, Sweden, 1852 & 1856 ................................................................................................. 20 21. Decher’s Planimeter, 1855.............................................................................................................................. 20 22. Stadler’s Hyperbole Planimeter, Austria, 1855 .............................................................................................. 21 23. Buniakovsky’s (Bunyakovsky) Planimeter, Russia, 1855 .............................................................................. 22 24. Dasnoy’s Calculateur-Rapporteur, Belgium, 1856 ......................................................................................... 23 25. Gangloff’s Planimeter, Bohemia, 1856 .......................................................................................................... 24 26. The Transformateur, the Netherlands, 1867. .................................................................................................. 25 27. Hofmann’s Instrument, Germany, 1874. ........................................................................................................ 25 28. Ljungström’s Cirkelplanimeter, Sweden, 1870 .............................................................................................. 26 29. Mechanised Harp Planimeters, Sweden, 1861 - ~1900 .................................................................................. 27 30. Kloth’s Hyperbolic Plates, Germany, 1883 .................................................................................................... 31 31. von Erichsen’s Planimeter, USA, 1884 .......................................................................................................... 31 32. Updated timeline of early area measurement instruments .............................................................................. 32 33. Bibliography ................................................................................................................................................... 35 1. The challenge to structure the topic In the Slide Rule GAZETTE 2021, one of us published a comprehensive overview [Zer2021-4] of early measuring instruments. In the meantime we have collected additional information for a second part on this topic. Tracing the development of these instruments in the 19th century presented many difficulties. So structuring this article was also a challenge. As a first approach, we tried to group the instruments according to the regions where they were invented and further improvements were made. This was particularly difficult because many of the instruments weren't widely used or even widely known (see, for example, the Swedish instruments mentioned below). However, we soon realised that for instruments that were more widespread, this regional categorisation no longer seemed to work. Another approach to categorisation was by the underlying principle of the instrument. Again, we felt that this approach did not work well because some devices have overlapping principles and it was not possible to make a clear distinction between the inventions. Furthermore, due to the large number of devices invented, we wanted to pay special attention to early area measurements that had not been described in detail in previous 61 Slide Rule GAZETTE Issue 23 Autumn 2023 articles such as [DreHae2009-1, DreHae2011-1, Zer2021-4]. For this reason, we have decided not to use this categorisation scheme, which would have required us to re-examine the instruments from the articles cited above. We have therefore chosen to present the inventions in chronological order, as summarized in the last section of this article. We will also try to provide information on the relationship between the various instrumental developments and their inventors. 2. Early usage of the term “Planimetry” Before we start our journey into history of early instruments used for area measurement, let's have a closer look at the definition of the word planimetry. Planimétrie (or planimetry) is the study of plane measurements, including angles, distances and, in the particular context of this paper, surface areas. An early work on the subject is “La Geometrie Pratique” subtitled “La Trigonometrie Theorique & Pratique, la Longimetrie, la Planimetrie, & la Stereometrie” by the French mathematician Jacques Ozanam (1640-1718). Ozanam not only described conventional methods for calculating areas, but also introduced a graphical geometric transformation method, which would not be supported by mathematical instruments until more than 100 years later. The French cartographer and fortifications engineer Allain Manesson-Mallet (1630-1706) treated the subject of geometric transformations more extensively in the richly illustrated 4-volume work “La Geometrie Pratique” of 1702, with many worked examples in the chapter on planimetrie [Mal1702]. Figure 1: Geometric transformations. Left: graphically transforming a trapezoid into a triangle of equal area [Oza1684], p. 209. Right: transforming a more complicated polygon into a triangle of equal area, ManessonMallet, p. 233 [Mal1702]. 3. Bryon’s Planimetre Graphique, France, 1814 and 1826 We will start our journey with the history of the early area measurements in the year 1814. Charles Henri Bryon, ingénieur géomètre en chef du cadastre, had been using the common method of dividing a polygon into triangles, measuring the height and base of each triangle, multiplying the products, adding the results and dividing by 2. Towards the end of 1809, aware of the advantage that a simple and rapid method could have over the known means, he began to experiment with the geometric transformation method to reduce the area of a polygon to the calculation of the area of a single triangle. We describe this approach in the section on Zaręba’s planimeter below. As he continued his research, he noticed that if he transformed the triangle with an equal in area to the polygon into another right-angled triangle of the same size, with a fixed base and a variable height, he would always have 100 for one of the triangle’s dimensions, i.e. for one of the multiplication factors, which could then be done without using a pencil or even tables. He also felt that by giving the base of the triangle a fixed length, it was necessary to measure the height with great accuracy, which was made difficult by the small scale of 1:2500 generally used for land registry plans. This led Bryon to choose a scale for calculating areas that was twice as large as that of the plan. To reduce an area to its real value, instead of multiplying the height of the triangle by half the base, it was necessary to multiply it only by 2. From this he concluded that, in order to always have 100 for one of the factors of the 62 Slide Rule GAZETTE Issue 23 Autumn 2023 product, it was necessary to give the triangles a base of 50 meters, a length which fitted most triangles in practice. In other words, when calculating the area of a polygon, a double transformation would simply reduce it to the measurement of a single line, which would have to be multiplied by 100. Figure 2: Bryon's Planimetre Graphique from [Bry1814] (Courtesy of L'Héritage des Ponts et Chaussées). Figure 3: Bryon's Planimetre Graphique variant from 1826 [Bry1826] (Courtesy of Biblioteca e Arquivo Histórico da Economia). To make the transformations easier, Bryon devised an instrument which he called a Planimètre Graphique and published a treatise on its use in 1814 [Bry1814]. However, it was necessary to mark the intersection of the parallels with the base with a pencil line on the base, and Bryon noted that there was a risk of confusing the last mark with one of the previous ones. This prompted Bryon to create a new model in 1826 that 63 Slide Rule GAZETTE Issue 23 Autumn 2023 avoided the need to mark the baseline, and without having to draw a single line other than that of the triangle equivalent to the polygon [Bry1826]. Apart from references to Bryon's treatises of 1814 and 1826 in various bibliographies, we couldn't find any other references to his planimeter in the literature. In fact, later inventors seem to have been unaware of Bryon's planimetre graphique, as it is never mentioned, leading us to the cautious conclusion that it must have been known and used only locally. 4. Gelinski’s Calculateur, France, 1815 Frederic Charles Gelinski, was a Polish nobleman born 10 August 1771 in Wilna, Poland. Fighting against the French during the French Revolutionary Wars he was taken prisoner in 1792 with his Austrian regiment. He remained in France and was admitted to the Ponts et Chaussées in Troyes, then, in 1812, to Holland, then controlled by France. Naturalized in 1814 he became head of the Cadastre in Angers [NN1815] and was sent in disgrace to Digne following the July Revolution of 1830. He died there on 18 October 1837 [NN1983]. Figure 4: Gelinski’s Calculateur from [Lec1846]. Gelinski devised an instrument that facilitates the transformation of an irregular polygon to its equivalent triangle by constructing parallel lines. The area can then easily be calculated from the resultant triangle. Gelinski described its usage in a pamphlet published in [Gel1815]. Gelinski’s pamphlet is listed in several bibliographies but we have been unable to locate a copy in any library. However, there is a ZVAB bookseller in France who has, or had, an undated handwritten and illustrated copy of Gelinki's treatise. Gelinski’s Calculateur was considered to be sufficiently useful to be treated in several popular 19 th century French surveying textbooks. One is by Foucard and Place [FouPla1838], another being by François Lecoy [Lec1846] and from the Netherlands in [NN1850-1] as well. 64 Slide Rule GAZETTE Issue 23 Autumn 2023 Although Gelinski may not have been aware of Bryon's planimeter, as he makes no mention of it, his instrument is similar in construction. Two years after Gelinski’s idea, Zaręba’s device would also embody the same approach. The geometrical principle is explained in the next section. 5. Zaręba’s (Zaremba) Planimeter, Poland, 1817 In his book [Zar1829] Zaręba1 summarized the existing approaches by Zobel [MulZob1815], Colberg [Col1825], Wagner [Wag1821], Possener [Pos1823] and Wießner [Wie1828] and evaluated all of them as insufficient. In his introduction2, Zaręba mentioned: “The methods of calculating the geometric plans, which have been practised until now, as tedious and time-consuming, have given me reason to look for other, more appropriate circumstances. Not having any languages other than my mother tongue, I could not make use of the inventions in this matter described in a foreign language; and therefore I did not dare make my own known to the public, remaining in constant uncertainty as to whether my methods were not a repetition of those described in the foreign language. Methods are not a repetition of those already known. While constantly engaged in the practice of geometric measurements and visiting various localities, I came across people to whom I had to explain my methods facilitating geometric calculations in order not only to satisfy their curiosity but also to gauge their opinion on the usefulness of the invention. I was able to compare and contrast such opinions, as I found more in favour of the usefulness of the invention than against; I did not, at the same time, oppose the insistence and assurances of good effect of the experts and experts in mathematical skill; and, above all, I took advantage of the assistance of His Excellency Duke Adam Voivode of Warsaw. Duke Adam Voivode Czartoryski, I dared to ask the competent Government Authorities to issue me with an official certificate of the novelty and usefulness of my invention, as well as for permission to publish the dissertation in print and dedicate it to my co-workers for their convenience. My requests received the desired result. I therefore grant to the co-operators both the invention for which I have been granted, and the confirmation of the Government which I have received, which will serve as a kind of prior assurance that my tool can be used to the advantage of my colleagues. My tool can be used profitably.” Figure 5: Zaręba’s approach: 1st iteration moving point 1 to 1'. Figure 6: 2nd iteration: Moving 2 to 2' - the 8polygon becomes a 7-polygon of identical size. Interestingly on page VII of [Zar1829], Zaręba mentions that on 11 th March 1817 he was granted the invention for 10 years for the whole Kingdom of Poland. Since the sovereign patent legislation operated in the Kingdom of Poland from 1817 to 1867 [Dol2018], Zaręba’s patent must be one of the first Polish patents. Therefore, although we don’t have any further evidence of the granted patent, we can date Zaręba’s planimeter to 1817. In addition, according to the note in [NN1829-3], Zaręba renewed his patent in 1829. Zaręba gives a detailed description of his invention. Although the method of graphically transforming a closed polygon into a triangle of equal area was known much earlier (see section 2 and [Mal1702]), the method is quite remarkable. We will now explain how it is done using Zaręba’s device. He transforms an arbitrary closed polygon with n corners into a polygon with n-1 corners (of the same area) by geometrically moving one corner to a horizontal base line. Again we emphasize that by the geometrical construction the n-1-polygon has 1 2 Sometimes Zaręba is called Zaremba, because of the Polish pronunciation of the name Zaręba. Translated by DeepL with some edits. 65 Slide Rule GAZETTE Issue 23 Autumn 2023 the same area as the original n-polygon. The same process is applied to the n-1-polygon resulting in an n-2polygon. Again the n-2-polygon is of identical size as the n-1-polygon, which is of same size as the original npolygon. This process continues until the construction results in a triangle (which is of the same area as the original n-polygon). Determining the area of the final triangle therefore gives the area of the original n-polygon. Figure5 shows an 8-polygon from [Gra1908-1] explaining Zaręba’s approach. Consider the closed polygon with 8 corners, which area is to be measured. A horizontal line is drawn crossing point 0. Next a diagonal line through points 0 and 2 is taken. Now this line is parallel shifted through point 1. This line intersects with the horizontal line at 1’. The core idea is that the triangles 012 and 01’2 have the same area; because both triangles have the same base 02 and the same height (the distance between the two parallel lines) As a consequence the closed 8-polygon 012345678 is of identical size to the polygon 01’2345678. The next iteration (Figure 6) starts with the polygon 01’2345678, the diagonal 1’3 is taken. This diagonal is parallel shifted through point 2, intersecting the horizontal line at 2’. Again the triangles 1’23 and 1’2’3 are of same size. Consequently the polygons 01’2345678 and 01’2’345678 have the same size. But, be aware, the polygon 01’2’345678 does not have 8 corners any more – there are only 7 corners left (since the points 0, 1’ and 2’ are laying on the horizontal line). Iterating the process for each point 4, 5, 6 and 7 reduces the number of polygon corners by one, finally leading to a triangle with identical size of the original polygon. The size of the final triangle can be determined by measuring the base and the height. With this knowledge we are now able to understand Zaręba’s device. It graphically supports determining the intersection points 1’, 2’,…. on the horizontal line. Figure 7: Zaręba's planimeter from [Zar1829]. Compare the similarity with Gelinski’s device in Figure 4. The device is placed below a figure to be measured and consists of a horizontal ruler carrying an unnumbered scale. Inside the ruler two sliding scales can be moved and afterwards fixed by screws. The lower sliding rule carries the index numbers for the unnumbered rule. The upper sliding rule provides a Vernier. On this upper sliding rule at point d a protractor (a and a’) is mounted. The rotatable arm a’ is used to be the diagonals (e.g. 02 in Figure 5, or 1’3 in Figure 6). After positioning the arm as such a diagonal, one moves the upper sliding rule horizontally, which realizes the parallel shift of the diagonal. This procedure is iteratively applied as above described. In the final step a rectangular triangle has been constructed, which area is easily determined with the help of the index numbers on the lower scale. For a more detailed description of the usage of Zaręba’s planimeter we refer to [Gra1908-1, Gra1908-2]. 66 Slide Rule GAZETTE Issue 23 Autumn 2023 The fact that the Polish inventions have been hardly known outside Poland has also been addressed in the Austrian journal on surveying by Grabowski [Gra1908-1, Gra1908-2]. In the article Grabowski refers to a discussion on the priority of the planimeter inventions of Gangloff and Schlesinger in [Mul1879]. In the referred article Müller allocated the priority of the invention to Gangloff based on his publication in 1856 [Gan1857]. Grabowski did not agree with this statement and pointed out that Gangloff's planimeter was identical to the planimeter invented by Jan Zaręba in 1829. There is only a small mechanical difference between the two instruments. Zaręba used locking screws to prevent unintentional rotation of the arm a'. Gangloff used a scale mounted on a glass plate for the same purpose. Grabowski noted that Gangloff's construction was more prone to errors than Zaręba's planimeter because of the reading through the glass plate. 6. Laur’s Olarithme, France, 1819 Géomètre Jean-Antoine Laur (born 1785) introduced a planimeter, which he called the Olarithme, in the first edition of his Géodésie Pratique, published in 1819. This popular French surveying textbook went through many editions, the 6th being published in 1855 [Lau1855] and a 2-volume German edition in 1857 [Lau1857-1, Lau1857-2]. Laur presented the Olarithme to the French Academy of Sciences in 1821. He continued to extol the merits of his Olarithme in his Notice sur les Planimètres published in 1859 [Lau1859]. Laur participated in the Great Exhibition of 1851 and received an honorable mention for his Olarithme. Favaro, quoting a report of the jury, gave the following description of the Olarithme Figure 8: Laur's Olarithme “Laur (France, Nr. 567) has exhibited a planimeter in which the area of a plane, supposed to consist of triangles, is first reduced, by geographical construction, to any right-angled triangle. On a rectangle of either gelatin paper or transparent horn are engraved a series of hyperbolas, such that the product of the ordinate and abscissa contained for each, shall change from hyperbola to hyperbola by a constant increment. The extremity of one side of the triangle being made to coincide with the common centre of all the hyperbolas, and that side 67 Slide Rule GAZETTE Issue 23 Autumn 2023 with the asymptote, the opposite angle will of necessity fall on one or between two adjacent ones of the hyperbolas, and its area (or its double) will be there-fore expressed by the ordinal number of the hyperbola, or the number of the lowest plus or proportional part for the distance between that and the angle, as compared with the interval between the two consecutive curves...” [Fav1873]. 7. Colberg’s Planimeter, Poland, 1820 Let us now look at the inventors in Poland. An overview of Polish planimeters had been published by Feliks Kucharzewski in 1902 [Kuc1902]. The Polish instruments have not spread much beyond the borders of the region. For this reason, there are very few articles on the devices and references to them outside the Polish language. A detailed description of Julius Colberg’s device has already been given in [DreHae2009-1], but without listing any corresponding references. Colberg’s original Polish [Col1824] and German [Col1825] publications give details of the invention. Christoph Julius Heinrich Colberg (also spelled Kolberg, or in Polish Krzysztof Juliusz Henryk Kolberg) was born on 7 July 1776 in Woldegk and died on 5 September 1831 in Warsaw. Colberg was aware of Zobel’s and Wagner’s planimeter and referred to their publications from 1815 and 1821. He left it up to others to compare the performance of Zobel’s and Wagner’s devices with his. To Figure 10: Area of area = base b * height h / 2. a triangle: Figure 9: Closed Polygon abcdefgh divided into six triangles ([Col1825] Tab. II). claim the date for his invention Colberg referred to an article in a Polish forestry journal from 1820, where he originally published his device [Col1820-1]. Educated at the Technical University of Berlin, Juliusz Colberg became Professor of Surveying at the Royal University of Warsaw in 1817. He was a teacher at a local forestry school and an active member of the Royal Society of Friends of Science in Warsaw. In 1820 he was awarded a Master of Fine Arts and a Doctor of Philosophy by the University of Warsaw for his work in cartography and the construction of his planimeter. He also translated Polish poetry. Colberg’s approach of determining areas of closed polygons is based on dividing the area into triangles The areas of these triangles are determined and summed up, leading to the total area enclosed by the polygon. To understand the approach we need to remember our trigonometry courses at school: The area of an arbitrary triangle is calculated by the base times the height divided by two. Instead of calculating the rectangle area b x h, Colberg applied the principle of rectangle squaring, i.e. the area of a rectangle b x h equals the area of a square with side length calculated by the geometric mean 𝑎 = √(𝑏 ∗ ℎ) Therefore the area of the triangle is (a * a) / 2. Colberg’s approach is to find the geometric mean of the base and height of any arbitrary triangle. With the geometric mean the area of the triangle can immediately be read from a corresponding multiplication table (Figure 11) which Colberg calls planimeter. The results in the table are in the old plot area units of “Morgen” and “Quadrat-Ruthen” (square rods), where 1 Morgen equals 180 square rods. In the multiplication lattice the large, bold numbers represent the Morgen and in the cells right to this number the additional square rods are given. A more detailed look at the multiplication table shows that the table consists of two parts, the upper lattice in Figure 11 and the lower lattice above the proportional circle. This lower lattice is the continuation of the upper lattice at its right end. So the complete lattice consists of nine horizontal lines plus one skew line at the bottom. Additionally there are 101 vertical lines (dividing the table into 100 columns). Looking up the result (a * a) / 2 for a given geometric mean “a”, one needs to look at corresponding cell. Moving 1 cell up increases the geometric mean “a” by 10 rods. Moving one cell to the right increases “a” 68 Slide Rule GAZETTE Issue 23 Autumn 2023 by 1/10 rods. Be aware moving from the first column to the last column leads to an increase of 100 * 1/10 rods = 10 rods. This is the reason for the skewed line starting at left at zero and ending at the right with a distance to the first horizontal row representing 10 rods. Figure 11: Colberg's multiplication table for squaring the geometric mean from [Col1825]. As an example, let us search the result for (a * a) / 2 for a = 53.2 rods. For this we need to look right of column 32 (32 / 10 rods) (be aware the first column represents 0/10 rods) and up above the 5 th row representing 50 rods (be aware the 1st row represents 0 rods). The corresponding table entry shows 155 (square rods) and the first large number to the left is a 7 (Morgen). Additionally, below the corresponding column a small 12 is noted indicating 12/100 square rods. In total the area of triangle with the geometric mean “a” equals to (a * a) / 2 = 53.2 x 53.2 /2 square rods = 1415 square rods = 7 Morgen 155.12 square rods. Although Colberg explained that the results of his planimeter (the multiplication table) would only be reliable if the device was made entirely of brass with precisely engraved lines. Printed on paper, the results of the planimeter would vary greatly due to changes in the temperature and humidity of the paper. After the description of above calculations, the geometric mean of the base and the height of triangles need to be determined. Colberg describes a geometric procedure using a reduction compass (see lower left in Figure 11), which physically reduces the distances between the tips of the longer compass arms to one half between the tips of the shorter compass arms. First Colberg puts the base and the height of the triangle on the horizontal line (see Figure 12). With the reduction compass he gets the arithmetic mean a = (b+h)/2 between the tips of the shorter compass arms. Taking the distance a from the endpoint b (or endpoint h) one compass tip is now in the centre between Figure 12: Geometric determination the two end points of b and h. Without changing the compass he takes a of geometric mean of h and b. circle around this centre point to determine the intersection with the orthogonal axes. The distance m between this intersection and horizontal axes is the geometric mean: 𝑚 = √(𝑏 ∗ ℎ). Taking this distance in the compass the value for m is read with the scale (see bottom right in Figure 11. This is followed by the multiplication described above. 8. Wießner’s mathematical instrument, Jena Germany, 1823 & 1828 From an early age Johann Gottfried Wießner had a special affinity for mathematics. He gives an account of his early years in the introduction of his treatise on parallel lines in 1833 [Wie1833]. In 1809, after 69 Slide Rule GAZETTE Issue 23 Autumn 2023 receiving the necessary training, he found employment as a geometer with the Royal Bavarian Land Survey. From 1815 to 1817 he participated as a geometer in the survey of the Duchy of Altenburg and for the next 12 years in the national survey of the Grand Duchy of Saxony-Weimar-Eisenach. In the 4 years before writing his treatise in 1833, in addition to his profession as a geometer, he gave private lessons in mathematics. From 1852 to 1866 Wießner was lecturer in map making at the Gesamt-Universität Jena. His year of birth and death are not known. At that time, the common method of calculating the area of a polygon was to divide it into triangles and add up the areas of each triangle. Having conceived of a more efficient method Wießner proceeded in the winter of 1819 to draw up a plan for his device and then commissioned Hofmechanikus Friedrich Körner [Wik2023-1] in Jena to fabricate it. Wießner introduced his mathematical instrument in a treatise “Beschreibung eines neuen mathematischen Instruments” published in [Wie1823]. It was constructed of hard wood, inlaid with brass rulers with divisions in Weimar foot measure. The triangle could be moved within the rectangular frame so that the abscissa and ordinate of each point in the polygon could be read on the rulers. The area can then be calculated from the coordinates obtained, as was originally proposed by Gauss in 1790 [Ams1907-1, Ams1907-2, Ell1925]. In the introduction to his treatise on his calculating clock in 1828, Wießner notes that his mathematical instrument was currently being manufactured in Berlin and used to enter data into astronomical charts. Figure 13: Wießner's mathematical instrument from [Wie1823]. Wießner was already familiar with Zobel’s and Müller’s planimeters when he published his treatise in 1823, and wishing to create an instrument more suitable for Saxe-Weimar, where the units of area were not decimal, he set about designing a calculating clock that would make area calculations easier. In the beginning of 1828 Wießner constructed a model of a calculating clock with wooden cogwheels, which he described in “Beschreibung einer mathematischen Rechnungsuhr” (Figure 14) published in the same year [Wie1828]. On the title page Wießner gives this concise description: “By the use of this clock the whole calculation is 70 Slide Rule GAZETTE Issue 23 Autumn 2023 accomplished, because it adds everything, divides every line by 2 and multiplies all figures, both triangles and trapeziums from 1 to 200 feet, or 20 Paris lines in height.” The fact that Wießner had to deal with non-decimal units of area meant that his clock had to be more complicated than other calculating devices. The Paris line is an old unit of length. 12 Paris lines are equal to one Paris inch (pouce) which is longer than the English inch. In 1799, the unit Paris line was used to determine the length of the standard meter, which is equal to 443.296 Paris lines. One Paris line is therefore equal to 2.2558 mm. With his calculating clock, Wießner was able to determine triangles and trapezes up to 20 Paris lines (4.5 cm) in height. Figure 14 shows the detailed construction. Fig. II within this shows the device with its three dials: a) the large dial is divided into 100 decimal feet (we assume Paris feet) (equals 1 square ruthe (Engl. rod) b) the dial divided by 35 ruthen (equals a quarter acre) c) the dial divided by 12 complete acre. Figure 14: Wießner's Rechenuhr (engl. calculating Universitätsbibliothek Tübingen, Signature Bc 114). clock) from [Wie1828] (Courtesy When dial c) makes one complete turn (12 acre), the dial b) will turn 48 times (48 quarter acre = 12 acre) and dial a) will turn 1680 times (48 quarter acre = 48 x 35 ruthen = 1680 ruthen). Fig. V in Figure 14 shows the gear construction for the corresponding transmissions. Behind the gear construction, Wießner designed a multiplication system based on two identical, but opposite multiplication cones (centre of Figure 14). The cones have a circumference of 120 Paris lines (270.7 mm) at the base (corresponding to a diameter of 8.62 cm) and 6 Paris lines (approx. 13.58 mm) at the top (corresponding to a diameter of approx.t 4.3 mm). Both cones are connected by a fine chain that runs around both cones. Positioning the chain around the bottom of the left cone and the top of the right cone gives a transmission ratio of 1:20. Moving the chain to the top of the left 71 Slide Rule GAZETTE Issue 23 Autumn 2023 cone and therefore at the bottom of the right cone gives a reverse transmission ratio of 20:1 is in use. Multiplication is achieved with the different cone ratios. It should be noted, that the cones described do not have teeth to carry the chain. The cones are hollow and have four slots on their surfaces from the bottom to the top. Inside the hollow cones there are two additional cones (Fig. IV in Figure 14). On the surface of the inner cones, a screw thread runs around. In this screw thread on both inner cones 4 small forks can run. These four forks are guided in the four slots of the outer cones. The chain is driven by the four forks of both cones. So the transmission ratio between the two cone systems can be changed by rotating the inner cones against the outer cones, causing the forks in the slots of the outer cones to move between the bottom and top of the two cone systems. In modern terms we would call this a cone chain transmission. To measure the area (see in Figure 14 at the left of Fig. III), the height of the trapezoids (distance between the parallel sides,) is entered with the knob (bottom of Fig. III), which causes the inner cones to rotate and thus changes the transmission. The length of the parallel lines of the trapezoids is measured with the other knob (bottom left in Fig. III). The final result is displayed on the dials on the front of the unit. Although Wießner encouraged others to copy his calculating clock, no copy is known to have been made, nor does the original Wießner calculating clock appear to have survived. This may have been due to changes in units of measurement that made it obsolete, or it may simply have been found to be impractical to use. In 1835 Wießner published a textbook on surveying entitled “Praktische Anweisung zur Feldmeßkunst“ [Wie1835]. In the chapter on calculating the area of land plots, he curiously makes no mention of his mathematical instrument or his calculating clock. 9. Tenner’s Planimeter, Germany, 1828 In the course of our research, another unknown (at least to the authors) instrument for measuring surface area turned up in the literature. In the German journal “Didastalia” [NN1829-2] a device invented by Tenner is mentioned. In [Ten1828] Tenner described how to determine plot areas avoiding any calculation. On the right in five adjacent plots (1 to 5) are shown. The first plot (ABcd) is divided by a diagonal into two separate triangles (ABc) and (Acd). Remember again Figure 10. The area of a triangle is calculated by the base times the height divided by two. Applied to the triangles in Tenner’s plot 1, the area of (ABc) calculates to distance Ae times distance Bc divided by 2 and the area (Acd) equals Ad times cf divided by 2. Therefore, area of plot 1 equals to ((Bc x Ae) + (Ad x cf)) / 2 Tenner avoided the arithmetical multiplications by performing graphical multiplications. For this, he drew a set of similar triangles (left in Figure 15 ). He put equidistant marks p, p’,… on the ordinate and drew lines through these marks and the point C. Note: p’ = 2 x p, p’’ = 3 x p, etc. Based on the map scale with the figure to be measured, the distance Ap represents a real distance in the field. This value has been written next to the mark p. Since p’,p’’,… are multiples of p, corresponding multiples of the real distance are note to the corresponding marks p’,p’’,… On this ordinate the areas of the plot triangles will be read. The abscissa is divided into equidistant portions, starting from 1 at point A, 2 in the middle between A and C and corresponding intermediate values in between. At corresponding abscissa marks vertical lines are drawn. The determination of the plot triangles one had to take the measured width of a plot (e.g. Bc) as the base, chose the corresponding abscissa mark. The triangle height (in this case Ae) is taken by a circle. Afterwards one end of the circle is put in the chosen abscissa mark and with the other end the distance on the corresponding vertical line is marked. A line from C through this point will cut the ordinate at certain point. The corresponding value on the ordinate (divided by 2) gives the area of the triangle. The division by 2, can already be considered in the design by scaling the ordinate values by a factor of two. Tenner called these kind of “multiplication graphs” as “Inhaltsmesser” (Engl. content meter). Obviously several graphs for different ranges of plot width and map scales are required and given in [Ten1828]. He asked the readers to draw these “Inhaltsmesser” by themselves according to their needs. At this point of his treatise Tenner proposed to combine many different ordinate scales leading to hyperbolic curves engraved on a glass plate. With this device no circle is required any more. The glass plate could be moved over the figures to be measured and the results could be read directly without taking any intermediate measurement. This is an early description of a simple hyperbolic table. 72 Slide Rule GAZETTE Issue 23 Autumn 2023 Figure 15: Tenner's approach from [Ten1828], courtesy Universitäts- und Landesbibliothek Darmstadt. Tenner also created a mechanical aid to help calculate the above. A rotatable scale, fixed in point C, can be used to read the multiplication results on the ordinate. To make it even more convenient, the results to be read are written on the circular arc (Figure 6 Figure 25 on the left the arc BD). From this point Tenner called his device a calculating disc and explained also how it can be used for divisions and determining square roots. The treatise did not show any device. Since Tenner had the intention, that the readers create devices by themselves and the fact that we did not find any further reference to his device or even his treatise, we assume that the instrument he described has not been successful in the market, at least not under his name. 10. Littmark’s Pollettcirkel, Sweden, 1831 & 1857 Fredrik Littmarck (1801-1876), land-surveyor, son of mining-surveyor Jonas Littmarck (1758-1847) [Eks1903, p.91]. At that time a common method in Sweden was to place a glass plate with an engraved rectangular grid over an enclosure and simply tally up the number of squares [And1876, p.223]. Littmarck “endeavoured to find a shortcut by which one could reach the goal more easily”, devised an instrument which he called the Pollettcirkel, and in 1831 published a pamphlet describing its use [Lit1831]. Similar in appearance to a beam compass, the Pollettcirkel consisted of a