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Veltmann's Device for Solving a System of Linear Equations

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Veltmann's Device for Solving a System of Linear Equations faculty of mathematics and natural sciences Veltmann’s device for solving a system of linear equations Bachelor Thesis Mathematics July 2012 Student: H.J. Veenstra First Supervisor: Prof. dr. J. Top Second Supervisor: dr. M.K. Camlibel Abstract In 1892 the `Deutsche Mathematiker Vereinigung' (German mathematician society) pub- lished a catalogue of (physico-)mathematical models, devices and instruments. In this catalogue we find the description of a device invented by professor W. Veltmann. His device could be used to solve a system of linear equations by using hydromechanics. In this thesis we tried to discover Veltmann's motives to build such a device. Hereafter we looked at the device itself. We found that Veltmann might have built this device for educational purposes. The de- vice works because of an equilibrium according to the law of the lever and an equilibrium between the buoyant force and the gravitational force. Veltmann's device works properly for inconsistent systems and systems without a unique solution. However, there are re- strictions that need to be posed on amounts of fluid that are added in order to make the device work properly. Contents 1 Introduction 4 2 Historical Background 5 2.1 Biography of Wilhelm Veltmann . .5 2.2 Walther Dyck's catalogue . .6 2.3 Why did Veltmann build his device? . .9 2.4 Similar devices . .9 3 The device for one equation 10 3.1 The description by Veltmann . 10 3.2 The appearance of the device . 14 3.3 Solving the equation . 16 3.4 Practicability . 17 4 Why the device works 18 4.1 Physical background . 18 4.2 Computational background . 22 4.3 The device in the general case . 25 5 Restrictions 28 5.1 Restrictions on ti ................................ 28 5.2 Precision . 31 5.3 Error reduction . 32 6 Properties 33 6.1 Systems without a unique solution . 33 6.2 Inconsistent systems . 35 7 Conclusion 38 A A list of Veltmann's publications 44 2 B Computations for one equation 47 B.1 Properties of the box . 49 B.2 Expressing sin(θ~) in terms of a1:0 and a1:1 .................. 50 B.3 Expressing the other variables in terms of a1:0 and a1:1 ........... 52 B.4 Calculating y0 and y1 and the solution of the linear equation . 54 C Computations for a system with more than one equation 57 C.1 Some equations for the general case . 58 C.2 Results for two equations . 60 3 Chapter 1 Introduction In 1892 the `Deutsche Mathematiker Vereinigung' (German mathematician society) pub- lished a catalogue of (physico-)mathematical models, devices and instruments [vD92]. This catalogue contains descriptions of all kinds of devices and geometric models, some of which still can be found in museums or in the archives of several universities. However, some other devices seem to have completely disappeared. In this catalogue we find the description of a device invented by professor W. Veltmann, hereafter called `Veltmann's device'. This device could be used to solve a system of linear equations, not in an alge- braic way but by using hydromechanics. In mathematics there are more prevailing methods to solve such systems, which raises the question: \Why would somebody invent and build such a device?". In order to answer this question we first look at the life of professor Veltmann. Who was he and what where the main subjects of his research? We will also look at the develop- ment of the previously mentioned catalogue. Once this historical background is explored, we will look at the device itself: \How is it used and why does it work?" To fully understand this, some physical principles are recalled. Subsequently a way to reduce the error in our solution is proposed. In the end the behaviour of this device in some special cases is investigated. What happens if we have a system without a unique solution? Or a system without a solution at all? This thesis would not have been possible without the guidance and support of many people. I want to thank my first supervisor, prof. dr. J. Top, who has managed to make me enthusiastic about an old, strange device. I also want to thank dr. M.K. Camlibel, for being my second supervisor and for reading this thesis during his holiday. Furthermore I want to thank my study mate, Lianne. Day after day we were working on our theses, motivating and helping each other through. I also want to thank my parents and my boyfriend, who had to listen to long stories about old devices every time I spoke to them. 4 Chapter 2 Historical Background In this chapter some information will be given about Wilhelm Veltmann, the inventor of the device at issue. Furthermore, background information is given about Walther Dyck and his catalogue. In the end we will look at devices that are similar to Veltmann's device. 2.1 Biography of Wilhelm Veltmann Wilhelm Veltmann was born on the 29th of December 1832 in Hagen-Bathey, Germany [HAN]. We have no information about his youth. The first record that we found was that of his first article, which appeared in `Astrono- mische Nachrichten' in 1870 [Vel70]. This article suggests that he then lived in Bonn. Based on the addresses in his publications we know where he lived in the subsequent years. In 1871 he lived in Wiedenbr¨uck [Vel71]. In 1873 he had moved to Holzminden, where he worked at the Baugewerkschule (a school for builders)[Vel73]. In 1875 he lived in D¨urenwhere he was a teacher at a secondary school [Vel75]. In 1877 he became a rector in Remagen [Vel77]. After 1877 his whereabouts are unknown for 5 years. In 1882 he published some articles while he was living in Frankenthal(Pfalz) [Vel82]. In 1883 or 1884 he became a teacher at the landwirtschaftlichen Akademie (Agricultural Academy) in Poppelsdorf-Bonn, which later became part of the `Rheinischen Friedrich- Wilhelms-Universit¨at'in Bonn (University of Bonn)[Vel84b]. In 1892 he became a profes- sor at this academy.[Vel92] On the 6th of March 1902 he died in Hagen-Bathey, his birthplace [HAN]. Wilhelm Veltmann wrote many papers. A list of these is found in the Appendix (A). 5 From this list it can be seen that in the first years of his scientific career Veltmann was mainly interested in astronomy and physics. Later he treats a lot of isolated subjects. One subject that keeps coming back however, is the theory of `Beobachtungsfehler'(observation errors) in inter alia [Vel92]. This cited article was published in the same year as Dyck's catalogue, in which the description of Veltmann's device is found. Six years earlier Velt- mann wrote an article about “Aufl¨osunglinearer Gleichungen" (the solution of linear equations) [Vel86]. It is a possibility that Veltmann wanted to tell his students something about iterative approximations of the solution of a linear system and the observation er- rors that thereby emerge. As an example of an iterative process, his device can be used. This theory is supported by the fact that Veltmann himself poses, in the description of his device, a way to reduce the observation errors by using an iterative process. There is however no real source for this theory, so it is just a speculation. 2.2 Walther Dyck's catalogue In 1892 the Deutsche Mathematiker Vereinigung decided to organize an exposition of mathematical and physico-mathematical, instruments and devices on the occasion of their annual meeting. This meeting was to be held in September 1892 in N¨urnberg, Germany [vD92, p. III]. The task of organizing the exposition was assigned to Walther Dyck, a professor at the `Technische Hochschule' (Technical University) in M¨unchen. Walther Franz Anton Dyck was born on the 6th of December 1856 in M¨unchen [zSW59, p. 210]. Here he went to high school, where Oskar Miller, who later founded the German Museum, was one of his friends. He studied engineering at the Technical University in M¨unchen. Dyck was much influenced by his contact with Felix Klein, a famous mathematician at this university [Has99, p. 7]. Klein was convinced of the usefulness of looking at geometri- cal models for educational purposes. These models were produced by his own students. Walther Dyck also produced some of these models. In 1879 he received his PhD as a student of Klein with a dissertation about Riemannian surfaces. After his graduation he became one of Klein's assistants. He worked on the theory of groups and was the first mathematician to define the abstract definition of a group. In 1884 Dyck became a professor in M¨unchen. He used a lot of the educational principles that he learned from Klein. He also made his students produce mathematical models and graphic representations of geometrical objects. 6 Dyck played an important role in the foundation of the Deutsche Mathematiker Vereini- gung in 1889. At their assembly of 1891 he suggested to organize an exposition in 1892 [Has99, p. 8]. He might have obtained this idea from a similar exposition in London in 1876. In the introduction to the catalogue which appeared on the occasion of the exposi- tion of 1892 he writes that there has been a great development in the use and usefulness of mathematical models. He says: \So erschien es naturgem¨ass,die Gelegenheit der diesj¨ahrigenVersammlung der Deutschen Mathematiker-Vereinigung, die gleichzeitig mit der Versammlung der Gesellschaft deut- scher Naturforscher und Aerzte in N¨urnberg tagen sollte, zu ben¨utzen,um ein zusam- menh¨angendesBild dieser Entwickelung vorzuf¨uhren."[vD92, p. IV] (So it came naturally, to use the occasion of this years meeting of the German Mathe- matical Society, which would take place simultaneously with the meeting of the Society of German Scientists and Physicians in N¨urnberg, to demonstrate a coherent picture of this development) A lot of institutes made devices and models available.
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