Analog Computers

Reference / Paper · 2023

Analog Computing, 2nd Edition

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A comprehensive 460-page academic textbook covering the history, theory, and practice of analog computing from mechanical devices (Antikythera mechanism, slide rules, differential analysers) through classic electronic analog computers to hybrid and digital differential analyser systems. The book addresses basic computing elements (op-amps, integrators, summers, multipliers, function generators), programming techniques including scaling and partial differential equations, and an extensive survey of real-world applications spanning aerospace, nuclear engineering, biology, economics, and the arts. A final chapter covers the decline of analog computing in the late 1970s and its anticipated renaissance in low-power computing, AI, and high-performance computing.

Manufacturer
Walter de Gruyter GmbH
Author
Bernd Ulmann
Year
2023
Type
Reference / Paper
Language
English
Learning track
general theory
Pages
460
Credit
Bernd Ulmann, Analog Computing, 2nd edition. Berlin/Boston: Walter de Gruyter GmbH, 2023. ISBN 978-3-11-078761-0. https://doi.org/10.1515/9783110787740-202
  • Walter de Gruyter GmbH
  • analog computer history and theory
  • programming and scaling
  • hybrid computers
  • applications

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Analog Computing, 2nd Edition

Bernd Ulmann Analog Computing Also of interest Analog and Hybrid Computer Programming Bernd Ulmann, 2020 ISBN 978-3-11-066207-8, e-ISBN 978-3-11-066220-7 Quantum Information Theory Concepts and Methods Joseph M. Renes, 2022 ISBN 978-3-11-057024-3, e-ISBN (PDF) 978-3-11-057025-0 Multi-level Mixed-Integer Optimization Parametric Programming Approach Styliani Avraamidou, Efstratios Pistikopoulos, 2022 ISBN 978-3-11-076030-9, e-ISBN (PDF) 978-3-11-076031-6 Automata Theory and Formal Languages Wladyslaw Homenda und Witold Pedrycz, 2022 ISBN 978-3-11-075227-4, e-ISBN (PDF) 978-3-11-075230-4 High Performance Parallel Runtimes Design and Implementation Michael Klemm, Jim Cownie, 2021 ISBN 978-3-11-063268-2, e-ISBN (PDF) 978-3-11-063272-9 Algorithms Design and Analysis Sushil C. Dimri, Preeti Malik, Mangey Ram, 2021 ISBN 978-3-11-069341-6, e-ISBN (PDF) 978-3-11-069360-7 Bernd Ulmann Analog Computing 2nd edition Mathematics Subject Classification 2010 Primary: 34-04, 35-04; Secondary: 92C45, 92D25, 34C28, 37D45 Author Prof. Dr. Bernd Ulmann Schwalbacher Str. 31 65307 Bad Schwalbach ulmann@analogparadigm.com ISBN 978-3-11-078761-0 e-ISBN (PDF) 978-3-11-078774-0 e-ISBN (EPUB) 978-3-11-078787-0 Library of Congress Control Number: 2022946448 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Cover image: Bernd Ulmann Printing and binding: CPI books GmbH, Leck www.degruyter.com To my beloved wife Rikka Acknowledgments This book would not have been possible without the support and help of many people. First of all, I would like to thank my wife Rikka Mitsam, who not only did a lot of proofreading and a terrific job in preparing many of the pen-and-ink drawings, but also never complained about being neglected although I lived the last months more or less in seclusion writing this book. I am particularly grateful for the support and help of Jens Breitenbach, who did a magnificent job at proofreading and provided many suggestions and improvements enhancing the text significantly. He also spotted many LATEX-sins of mine, thus improving the overall appearance of this book considerably. In addition to that, I would like to thank Benjamin Barnickel, Arne Charlet, Daniela Koch, Theresa Szczepanski, Dr. Reinhard Steffens and Francis Massen for their invaluable help in proofreading. Additionally, I would like to thank Tore Sinding Bekkedal, who took the picture shown in figure 8.35, Tim Robinson, who built the incredible MeccanoDifferential Analyser shown in figure 2.22, Tibor Florestan Pluto, who took the photo shown on the title page, Robert Limes, who took the picture shown in figure 9.3, and Bruce Baker, who donated the pictures shown in figures 13.60, 13.62, and 13.63, for their permissions to use the aforementioned pictures. Without the continuous encouragement of Dr. habil. Karl Schlagenhauf and his invaluable suggestions this book might very well not exist at all. Last but not least, I would like to thank Prof. Dr. Wolfgang Giloi, Prof. Dr. Rudolf Lauber, Dr. Adolf Kley, Prof. Dr. Günter Meyer-Brötz and Manfred Klittich for sharing their memories of the early days of analog computing at Telefunken, etc. Furthermore, Jeroen Brinkmann introduced me to the “Hammer Computer” and Bert Brouwer offered many the insights into solving partial differential equations in conjunction with heat transfer problems. This book was typeset with LATEX. Most schematics were drawn using EAGLE and Arno Jacob’s wonderful analog computing symbol library, other vector graphics were created with xfig. Registered names, trademarks, designations, etc., used in this book, even when not specifically marked as such, are not to be considered unprotected by law. https://doi.org/10.1515/9783110787740-202 Preface to the 2nd edition This second edition of “Analog Computing” would not exist if it wasn’t for Dr. Damiano Sacco, acquisition editor at DeGruyter, who asked me to prepare a new edition of “Analog Computing”, which has established itself as a standard textbook since it was first published in 2013. A lot has happened since then in the world of analog computing. First of all many “new” historic sources, papers, and photographs have become available since 2013. Accordingly, the bibliography of this 2nd edition has been expanded by more then 200 additional sources. Second, and even more importantly, interest in analog computing has grown enormously. This computing paradigm is about to change the world of computing in the near future with a plethora of interesting and commercially important applications ranging from low power computing as required for medical implants to high performance computing, artificial intelligence, and many, many more. It seems even plausible that analog computers might be able to do things typically ascribed to quantum computers while being much simpler to implement, run, and program. Bringing back analog computers in much more advanced forms than their historic ancestors will change the world of computing drastically and forever. They will not replace the ubiquitous stored-program digital computers but they will complement them, thus making it possible to solve problems that are currently out of reach for standalone digital computers. The author is especially indebted to Dr. Chris Giles for many valuable discussions and his meticulous proofreading of this 2nd edition and to Nicole Matje for her valuable corrections and suggestions. The author would also like to thank Achim Dassow for information about the QK-329 beam-deflection tube and additional information on early multiplication devices. Rainer Glaschick provided much background information on early differential analysers and on the hyperbolic field tube. Oliver Bach also proofread the book and took special care of the bibliography making sure that it is consistent and correct. Prof. Dr. Dirk Killat contributed figure 12.9. The new cover picture was taken by Tibor Florestan Pluto and shows the author with a GTE-22 analog computer from the late 1960s. https://doi.org/10.1515/9783110787740-203 Contents Acknowledgments VII Preface to the 2nd edition IX 1 1.1 1.2 1.3 Introduction 1 Outline 1 The notion of analog computing Direct and indirect analogies 4 2 2.1 2.2 2.3 2.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.6 2.7 2.8 Mechanical analog computers 9 Astrolabes 9 The Antikythera mechanism 9 Slide rules 11 Planimeters 13 Mechanical computing elements 17 Function generation 18 Differential gears 20 Integrators 21 Multipliers 24 Harmonic synthesizers and analysers 25 Mechanical fire control systems 29 Differential analysers 32 3 3.1 3.1.1 3.1.2 3.2 3.3 3.4 3.5 41 The first electronic analog computers Helmut Hoelzer 41 The “Mischgerät” 42 Hoelzer’s analog computer 48 George A. Philbrick’s Polyphemus 57 Electronic fire control systems 61 MIT 67 The Caltech Computer 68 4 4.1 4.1.1 4.1.2 4.2 4.3 4.4 Basic computing elements 73 Operational amplifiers 73 Early operational amplifiers 77 Drift stabilisation 80 Summers 85 Integrators 89 Coefficient potentiometers 92 2 XII Contents 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5 4.6.6 4.6.7 4.6.8 4.7 4.8 4.9 4.10 4.11 4.12 4.13 97 Function generators Servo function generators 97 Curve followers 98 Photoformers 99 Varistor function generators 100 Diode function generators 100 Inverse functions 102 Functions of two variables 103 Multiplication 105 Servo multipliers 105 Crossed-fields electron-beam multiplier Hyperbolic field multiplier 107 Other multiplication tubes 108 Time division multipliers 109 Logarithmic multipliers 110 Quarter square multipliers 111 Other multiplication schemes 114 Division and square root 114 Comparators 115 Limiters 116 Resolvers 117 Time delay 117 Random noise generators 120 Output devices 121 5 5.1 5.2 5.3 5.4 5.5 5.6 The anatomy of a classic analog computer Analog patch panel 123 Function generators 124 125 Digital patch panel and controls Readout 127 Control 129 Performing a computation 131 6 6.1 6.2 6.3 6.4 6.5 6.6 Some typical analog computers 133 Telefunken RA 1 133 GAP/R analog computers 136 EAI 231R 138 Early transistorised systems 141 Later analog computers 147 THE ANALOG THING 150 106 123 Contents 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.5 151 Programming Basic approach 151 Kelvin’s feedback technique 153 Substitution method 155 Partial differential equations 157 Quotient of differences 158 Separation of variables 160 Scaling 162 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 Programming examples 165 Solving ÿ + ω 2 y = 0 165 Sweep generator 166 Mass-spring-damper system 168 Predator and prey 172 Simulation of an epidemic 174 Bouncing ball 176 Car suspension 181 Lorenz attractor 187 Mathieu’s equation 190 Projection of rotating bodies 194 Conformal mapping 196 9 9.1 9.2 9.3 Hybrid computers 201 Systems 201 Programming 205 Example 206 10 Digital differential analysers 209 210 10.1 Basic computing elements 10.1.1 Integrators 210 10.1.2 Servos 213 10.1.3 Summers 214 10.1.4 Additional elements 214 10.2 Programming examples 215 10.3 Problems 216 10.4 Systems 217 10.4.1 MADDIDA 217 10.4.2 Bendix D-12 220 10.4.3 CORSAIR 224 10.4.4 TRICE 225 XIII XIV Contents 229 11 Stochastic computing 12 12.1 12.2 12.3 12.4 Simulation of analog computers 233 Basics 234 DDA programming system for the IBM 7074 CSMP 237 Modern approaches 240 13 Applications 243 13.1 Mathematics 243 13.1.1 Differential equations 243 13.1.2 Integral equations 244 13.1.3 Roots of polynomials 247 13.1.4 Orthogonal functions 247 13.1.5 Linear algebra 248 13.1.6 Eigenvalues and -vectors 249 13.1.7 Fourier synthesis and analysis 250 13.1.8 Random processes and Monte-Carlo simulations 13.1.9 Optimisation and operational research 252 13.1.10 Display of complex shapes 253 13.2 Physics 253 13.2.1 Orbit calculations 255 13.2.2 Particle trajectories and plasma physics 255 13.2.3 Optics 258 13.2.4 Heat-transfer 258 13.2.5 Fallout prediction 262 13.2.6 Semiconductor research 262 13.2.7 Ferromagnetic films 264 264 13.3 Chemistry 13.3.1 Reaction kinetics 264 13.3.2 Quantum chemistry 265 13.4 Mechanics and engineering 266 13.4.1 Vibrations 267 13.4.2 Shock absorbers 268 13.4.3 Earthquake simulation 268 13.4.4 Rotating systems and gears 269 13.4.5 Compressors 270 13.4.6 Crank mechanisms and linkages 271 13.4.7 Non-destructive testing 272 13.4.8 Ductile deformation 272 13.4.9 Pneumatic and hydraulic systems 273 13.4.10 Control of machine tools 276 234 251 Contents 13.4.11 13.5 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.6.5 13.7 13.7.1 13.7.2 13.7.3 13.7.4 13.7.5 13.7.6 13.7.7 13.7.8 13.7.9 13.8 13.8.1 13.8.2 13.8.3 13.9 13.10 13.10.1 13.10.2 13.10.3 13.10.4 13.10.5 13.10.6 13.10.7 13.10.8 13.11 13.11.1 13.11.2 13.11.3 13.11.4 13.11.5 13.12 13.12.1 13.12.2 13.12.3 277 Servo systems Colour matching 278 Nuclear technology 278 Research 279 Reactor/neutron kinetics 280 Training 281 Control 282 Enrichment 283 Biology and medicine 284 Ecosystems 284 Metabolism research 285 Cardiovascular systems 285 Closed loop control studies 286 Neurophysiology 286 Epidemiology 288 Aerospace medicine 289 Locomotor systems 289 Dosimetry 290 Geology and marine science 290 Oil and gas reservoirs 290 Seismology 291 Ray tracing 292 Economics 292 Power engineering 295 Generators 295 Transformers 296 Power inverters and rectifiers 296 Transmission lines 297 297 Frequency control Power grid simulation 298 Power station simulation 300 Dispatch computers 301 Electronics and telecommunications 301 Circuit simulation 301 Frequency response 303 Filter design 304 Modulators and demodulators 304 Antenna and radar systems 305 Automation 305 Data processing 306 Correlation analysis 306 Closed loop control and servo systems 307 XV XVI Contents 13.12.4 13.12.5 13.13 13.13.1 307 Sampling systems Embedded systems 308 Process engineering 308 Mixing tanks, heat exchangers, evaporators, and distillation columns 309 Adaptive control 311 Parameter determination and optimisation 311 Plant startup simulation 312 Transport systems 313 Automotive engineering 313 Railway vehicles 317 Hovercrafts and Maglevs 317 Nautics 318 Aeronautical engineering 320 Landing gears 321 Aircraft arresting gear systems 323 Jet engines 323 Helicopters 323 Flutter simulations 324 Flight simulation 325 Airborne simulators 334 Guidance and control 336 Miscellaneous 338 Rocketry 338 Rocket motor simulation 338 Rocket simulation 339 Real-time data analysis 343 Spacecraft manoeuvres 343 345 Mercury, Gemini, and Apollo Military applications 347 Education 348 Arts, entertainment, and music 349 Arts 349 Entertainment 352 Music 354 Analog computer centers 354 13.13.2 13.13.3 13.13.4 13.14 13.14.1 13.14.2 13.14.3 13.14.4 13.15 13.15.1 13.15.2 13.15.3 13.15.4 13.15.5 13.15.6 13.15.7 13.15.8 13.15.9 13.16 13.16.1 13.16.2 13.16.3 13.16.4 13.16.5 13.17 13.18 13.19 13.19.1 13.19.2 13.19.3 13.20 14 14.1 14.2 14.3 Future and opportunities Challenges 359 Applications 361 Recent work 362 357 Contents Bibliography Index 427 365 XVII “An analog computer is a thing of beauty and a joy forever.”1 1 John H. McLeod, Suzette McLeod, “The Simulation Council Newsletter”, in Instruments and Automation, Vol. 31, March 1958, p. 488. 1 Introduction 1.1 Outline A book on analog computing and analog computers? You might ask: Isn’t that 60 years late? No, it isn’t – although the beautiful analog computers of the past are long since history and only few have been preserved in museum collections, the idea of analog computing is still a marvel of elegance and the following chapters will show that it has a bright and fruitful future. The intention of this book is twofold: It gives a comprehensive description of the history and technology of classic analog computing but also shows the particular strengths of the analog computation paradigm, which, combined with current state of the art digital circuitry, will find applications in areas such as low power computing, high performance computing and maybe most importantly in artificial intelligence (AI ) and many other fields. The following chapters first introduce the notion of analog computing before describing the early development of analog computers starting with mechanical analog computers like the Antikythera mechanism, which was built around 100 B.C., and ending with the first analog electronic1 analog computers developed by Helmut Hoelzer in Germany and George A. Philbrick in the United States. Next, the basic elements of a typical analog computer are described followed by two chapters showing the anatomy of typical analog computers, ranging from classic systems to more recent implementations, and showing examples of some systems. The next chapter gives an introduction to analog computer programming2 followed by a number of practical programming examples ranging from the solution of simple differential equations to the simulation of more complex and non-linear systems. Hybrid computers (analog computers coupled with stored-program digital computers) are covered in the next chapter. This is followed by a treatment of Digital Differential Analysers (digital implementations of analog computers), a chapter on stochastic computing, and a chapter on the simulation of analog computers on classic stored-program digital computers. The next chapter covers a plethora of classic and current applications of analog computing. 1 The notion of an analog electronic analog computer may look like a pleonasm, which it is not, as the following section will show. 2 A much more in-depth treatment of analog and hybrid computer programming can be found in [Ulmann 2020/1]. https://doi.org/10.1515/9783110787740-001 2 1 Introduction The last chapter of this book covers the decline of analog computing in the late 1970s/early 1980s and the potentially bright future of analog computing in the 21st century. 1.2 The notion of analog computing First of all it should be noted that the common distinction between digital and analog computers, based on the way values are represented, is not correct. It is often said that digital computers differ from analog computers by their way of representing numbers as sequences of bits (binary digits), while electronic analog computers work with continuous voltages or currents to represent variables. This erroneous view has even found its way into some encyclopedias. Apart from the fact that even voltages or currents are not really continuous – eventually an operation like integration boils down to charging a capacitor with discrete electron charges – some analog computers have used a bit-wise value representation and have been implemented using purely digital elements. If the type of values used in a computation – discrete versus continuous – is not the distinguishing feature, what else could be used to differentiate between digital and analog computers? It turns out that the difference is to be found in the structure of these two classes of machines: In our modern sense of the word, a digital computer’s constituent elements have a fixed structure and it solves problems by executing a sequence (or sequences) of instructions that implement an algorithm. These instructions are read from some kind of memory, thus, a better term for this kind of computing machine would be stored-program digital computer since this describes both features of such a machine: Its ability to execute instructions fetched from a memory subsystem and working with numbers that are represented as streams of digits.3 An analog computer on the other hand is based on a completely different paradigm: Its internal structure is not fixed – in fact, a problem is solved on such a machine by changing its structure in a suitable way to generate a model, an analog of the problem.4 This analog is then used to analyse or simulate the problem to be solved.5 Thus, the structure of an analog computer that has been set up to tackle a specific problem represents the problem itself while a stored-program digital 3 Today these numbers are normally represented by binary digits, bits for short. 4 [Tse et al. 1964, p. 333] characterized these analogs or analogies as follows: “The term ‘analogy’ is defined to mean similarity of relation without identity.” 5 The path from analogy-making to modelling of a problem is treated comprehensivly by [Care 2008]. 1.2 The notion of analog computing 3 Fig. 1.1. Comparison of the basic structure of stored-program digital computers and analog computers (see [Truitt et al. 1960, p. 1-40] and [Truitt et al. 1960, p. 1-41]) computer keeps its structure and only its controlling program changes. This is summarized by Charlesworth6 as follows: “An analogue computer is a piece of equipment whose component parts can be arranged to satisfy a given set of equations, usually simultaneous ordinary differential equations.” Similarly [Berkeley et al. 1956, p. 75] states that “[a]nalog computers, as the name is intended to imply, compute by means of setups that are analog of the problems to be solved.” Figure 1.1 shows this basic difference in architecture and operation between digital and analog computers. Consequently, it is perfectly possible to build digital analog computers and this has been done in several ways.7 In fact such machines may play a substantial role in the future when high precision is of the utmost importance and energy efficiency is a secondary consideration. Employing the techniques of building models, analogs of problems to be solved or analysed, which have been developed through more than 50 years in the context of our current digital technology, can and will lead to systems with exceptional computational power as well as low power consumption. 6 See [Charlesworth et al. 1974, p. xi]. 7 Cf. sections 10 and 13.15.8. 4 1 Introduction Stored-program control Setting up an analog Basic technology Analog electronic Digital electronic stored-program N/A (memory programmed) digital computer traditional digital analog electronic differential analog computer analyser Table 1.1. Types of computing machines based on an analog/digital electronic implementation with control based on either a stored-program concept or the implementation of an analog. Table 1.1 shows the four basic possible combinations of analog/digital implementation technology and stored-program control vs. setting up an analog. Of these combinations only three are of practical interest: 1. The modern stored-program digital computer, 2. the traditional analog electronic analog computer, which will be called an analog computer for simplicity in the following text, and, finally, 3. the Digital Differential Analyser,8 which will be described in more detail in chapter 10.9 1.3 Direct and indirect analogies When talking about analogies, it is necessary to distinguish between direct and indirect analogies. These two terms describe two extremes of abstraction levels in building analogies. In the strict sense of the word direct analogies are models that are based on the same physical principles as the underlying problem to be solved just with a different scaling regarding size or time of a simulation. Well-known examples of such direct analogies are the determination of minimal surfaces using soap films,10 the evaluation of tensile structures as they are used for roof structures 8 DDA for short. 9 It will be shown that DDAs are more capable machines than traditional analog computers since they can deal well with the highly important class of partial differential equations, which analog computers can do only with considerable difficulty and often only by means of discretisation. 10 See [Bild der Wissenschaft 1970] for examples. 1.3 Direct and indirect analogies 5 like the one built for the Olympic stadium in Munich,11 wind tunnel models for the evaluation of aerodynamic properties of aircraft and rockets and many more. Other direct analogs employed electrolytic tanks. These are reservoirs filled with an electrolytic liquid with embedded electrodes to generate a desired potential distribution within the liquid. Using a two- or three-dimensional sensor carriage, much like an xy-plotter, the potential at any given coordinate within the tank can be determined. Such electrolytic tanks were widely used to solve problems in nuclear engineering, etc. Over time the notion of direct analog analogies was also applied to computers consisting of networks of passive electronic components such as resistors, capacitors and inductors. [Paschkis et al. 1968, p. 5] defines a direct analog computer as being “based on the identity of the equations describing two or more systems and carrying out measurements on that system which appears most convenient for that purpose [. . . ] In the direct analog, there is a one-to-one relationship between the (passive) components and the physical properties of the several parts of the prime system.” Due to their very nature such direct analogs are not very versatile and were often built and employed for a highly specific purpose.12 Figure 1.2 shows a wonderful example for a direct analogy, a string-weight model used by Antoni Gaudí13 during the design and construction of the the Colónia Güell church. This model was built in 1908 at a scale of 1:10 with the weights scaled down by a factor of 10−4 . Such models are hanging down and simulate the pressure forces acting on pillars and columns by corresponding tensile forces through the maze of strings with their attached weights.14 In contrast, indirect analogies exhibit a much higher degree of abstraction and are thus much more versatile. The – mostly analog electronic – analog computers used for setting up indirect analogies are truly universal machines and cover a wide range of possible applications.15 This higher level of abstraction makes the programming of this class of machines quite challenging since their setup does not 11 The roof structure of the Olympic stadium in Munich was modelled to a large extent using curtain net lace as well as soap bubbles for determining the structure of single roof tiles. 12 More detailed information about this basic class of analogs can be found in [Jackson 1960, pp. 319 ff.], [Paschkis et al. 1968], [Master et al. 1955], [Larrowe 1955] and [Karplus 1958]. 13 06/25/1852–06/10/1926 14 See [Krämer 1989] for more details on this particular model. This approach was by no means new even in Gaudí’s time. See [Havil 2019, pp. 173 ff.] for a mathematical and historic perspective. 15 One of the earliest publications on the use of electronic analogs to simulate mechanical and acoustical systems was [Olson 1943]. 6 1 Introduction Fig. 1.2. Scale model for the Colonia Güell church bear any direct resemblance of the problem to be solved. Therefore a thorough mathematical description of the basic problem is required as a precondition for programming an indirect analog computer,16 as in the case of our modern storedprogram digital computers. Nevertheless, the level of abstraction required for the successful application of analog computers is still relatively small compared with the algorithmic approach of stored-program digital computers. Last but not least, analog computers, be they direct or indirect, are models. Due to the fact that analog computers work by acting as a model for a given problem that is represented by direct or indirect means, the amount of circuitry necessary for a simulation is determined by the complexity of the underlying problem. Accordingly, analog computers are not capable of the trade off between time to solution on the one hand and complexity of the underlying problem on the other that is characteristic of stored-program digital computers. This is both a 16 Direct analogs can also be employed in cases where no complete mathematical description of the problem to be solved exists – this may be caused by a principle lack of understanding or by the sheer complexity of the underlying problem. So in some cases direct analogs may even be employed today with success. 1.3 Direct and indirect analogies 7 curse and a blessing: The curse being that an analog computer consisting of a given number of computing elements cannot solve a problem that requires more computing elements to be implemented. The blessing is that the time to solution on an analog computer is more or less constant and is not related to the size of the underlying problem. Thus, large problems require large analog computers regardless of the acceptable time to solution – some classic problem areas, especially those found in aerospace and applications in the chemical industry, required well over 1 000 computing elements resulting in substantial, if not giant, analog computers. In addition to this, a stored-program computer can always exchange compute time for precision – something an analog computer also cannot normally do.17 The precision of an analog computer is given by its particular implementation and typically does not exceed about three to four decimal places for the variables involved in a computation. 17 This does not hold true for digital differential analysers, cf. section 10. 2 Mechanical analog computers The earliest analog computers were mechanical in their very nature but were far from being simple. In fact many mechanical analog computers were successfully employed to tackle complex problems ranging from peaceful tide computations to war-time applications like bomb trajectories, fire control, etc. The following sections give a short overview of the era of mechanical analog computers without going too much into detail since mechanical analogs will serve just as a prelude to this book’s main theme of electronic analog computers. 2.1 Astrolabes As early as about 150 B.C. the basics of astrolabes were developed. Such devices are basically inclinometers with some additional mechanics to model basic properties of spherical astronomy. Astrolabes are based on the apparent motion of celestial bodies, i. e., the observation that the paths described by stars in the sky are basically circles. Thus, the most common type of astrolabe is the planispheric astrolabe, developed in medieval times, which projects the firmament to the equatorial plane. Using such an instrument it is possible to determine the position of some celestial bodies at a given time. As a navigational tool the planispheric astrolabe is far too imprecise. Nevertheless, it has been used to roughly located the stars for getting navigational fixes. Detailed information about astrolabes can be found in [Dodd 1969] and [J. E. Morrison 2007]. 2.2 The Antikythera mechanism More than 120 years ago, in 1900, sponge divers found a lump of corroded gears in a Roman ship wreck, which carried treasures from Greece dating back to about 100 B. C. It turned out that these were the remains of one of the most complicated mechanical and mathematical devices ever. Due to its location near the Greek island Antikythera (AntikÔjhra) this impressive machine became known as the Antikythera mechanism. Figure 2.1 shows the main fragment of this early analog computer in its current state of preservation.1 1 Picture taken by Tilemahos Efthimiadis, protected by the Creative Commons Attribution 2.0 Generic license. https://doi.org/10.1515/9783110787740-002 10 2 Mechanical analog computers Fig. 2.1. Main fragment of the Antikythera mechanism as displayed in the National Archaeological Museum, Athens, Greece Intrigued by this find, Derek de Solla Price2 started investigating the inner workings of this device and summarized his astonishing discoveries as follows:3 “It is a bit frightening to know that just before the fall of their great civilization the ancient Greeks had come so close to our age, not only in their thought, but also in their scientific technology.” It turned out that the Antikythera mechanism was ahead of its time by at least 1 000 years. It is of such high complexity that recent research using modern X-ray tomography techniques, etc.4 continues to deliver new insights. New capabilities and details were discovered as late as in 2021/2022.5 The device modelled the movements of several celestial bodies, even taking into account various anomalies, 2 01/22/1922–09/03/1983 3 See [Freeth 2008, p. 7]. 4 Cf. http://www.antikythera-mechanism.gr/. 5 See [Freeth et al. 2021] and [Freeth 2022]. 2.3 Slide rules 11 which required differential gears6 and much more complicated epicyclic gearing. This astonishing complexity of the Antikythera mechanism led Mike Edmunds7 to the following statement: “Nothing as sophisticated and complex is known for another thousand years. This machine rewrites the history of technology. It is a witness to a revolution in human thought.” This intricate mechanism allowed the calculation of sun and moon positions at given dates and phases of the moon as well as the prediction of solar and lunar eclipses.8 The implementation of these functions required more than 30 gears, manufactured with extraordinary precision.9 2.3 Slide rules One of the most common, simplest and well-known analog computers is the slide rule,10 which comes in basically two configurations, linear, circular, and helical.11 The basic idea of a slide rule is to reduce the problem of multiplication and division to that of addition and subtraction by employing logarithmically divided scales that may be displaced accordingly to each other in a lateral direction. The analog setup in this case is to mechanize the relation log(ab) = log(a) + log(b) by using two logarithmic scales. Following the development of the logarithm by John Napier12 and Henry Briggs,13 who introduced the base 10 for logarithms, it was William Oughtred,14 who described the principle of the slide rule in his seminal two publications The Circles of Proportion, and the Horizontall Instru- 6 Prior to this discovery differential gears were thought to have been invented in medieval times. 7 See [Freeth 2008, p. 9]. 8 There are still arguments whether the mechanism also featured indicators for the display of planet positions. 9 A wealth of information about this device may be found in [de Solla Price 1974], [Freeth 2008] and [McCarthy 2009]. 10 Also known as a slipstick. 11 Additional precision is achieved by this type of slide rule by wrapping extended scales along a helical path. The downside of this is that such slide rules typically feature only two scales. 12 1550–04/03/1617 13 February 1561–01/26/1630 14 03/05/1574–06/30/1660 12 2 Mechanical analog computers Fig. 2.2. Typical scale (probably 18th century) ment and Two rulers of proportion.15 An early scale is shown in figure 2.2. Working with such a scale required a divider to transfer lengths from one of its engraved scales to another, a tedious and error-prone process that was greatly simplified by the introduction of a slider containing several scales, which led to the then ubiquitous slide rule. Figure 2.3 shows one of the last, most complex and most versatile slide rules ever built, a Faber Castell 2/83N. Its three main parts are clearly visible: – – – The body, which consists of the top and bottom stator or stock. The two stators are held together by two end braces or end brackets. The center slide, which can be moved laterally with respect to the body. The cursor, which slides in grooves of the body. While simple slide rules only feature a couple of scales, complex ones as the 2/83N have up to 30 and more scales, which implement functions far beyond multiplication and division.16 Using these scales, trigonometric functions, exponentiation, etc., can be evaluated. Until pocket calculators took over in the 1970s,17 slide rules were as widely used as they were centuries ago. The following quotation from Josef Vojtěch Sedláček18 shows this quite strikingly:19 15 A comprehensive history of the slide rule may be found in [Cajori 1994] and [Jezierski 2000]. A great introduction to the application and use of slide rules is given in [Hume et al. 2005]. 16 If all scales are just on one side of the body, a slide rule is called simplex. If scales are found on both sides of the body, it is a duplex slide rule. In this case the cursor is also used to transfer partial results from one side of the body to the other. 17 Early pocket calculators such as the Hewlett Packard HP-35 or some models made by Texas Instruments like the SR-10, etc., were explicitly marketed as electronic slide rules. The fixed number format featured by early HP calculators that was often set to display only 2 or 4 decimal places also was a reverence for the slide rule. 18 02/24/1785–02/02/1836 19 Cf. [Jezierski 2000, p. 16]. 2.4 Planimeters 13 cursor center slide body Fig. 2.3. Faber Castell slide rule model 2/83N “It is said that the use of the slide rule in England is so widespread that no tailor makes a pair of trousers without including a pocket just for carrying a ‘sliding rule’. During such a time, it is difficult to understand why the slide rule does not enjoy such well-deserved recognition in our own country.” Slide rules were essential tools for the scientific and technological progress of the last three centuries ranging from mathematics, civil engineering, commercial applications, electronics, chemistry, life sciences, etc., up to applications in aerospace technology.20,21 Although they were rendered more or less obsolete by pocket calculators 50 years ago, there are still areas of application where slide rules are employed regularly. For example, many aviators still use a flight computer like the E-6B, a special form of a circular slide rule, that allows the calculation of ground speed, i. e., the speed of an aircraft corrected for wind effects, and many other crucial parameters.22 Figure 2.4 shows a strange special purpose circular slide rule that was deployed in large amounts during the Cold War – a Nuclear Weapon Effects Computer, which allowed rough estimates of fatalities and damage should a nuclear air burst occur. 2.4 Planimeters Planimeters are fascinating instruments. Their purpose, most aptly described by [Henrici 1894, p. 497], is the following: 20 In fact Buzz Aldrin (01/20/1930–) carried a Picket slide rule on the Apollo-11 mission, which was sold on September 20th , 2007 for $77,675. 21 [Kaufmann et al. 1955] describes some interesting electronic circuits, most of which requiring only passive components such as potentiometers and resistors, to implement an electronic slide rule. 22 Apart from the ease of use, such specialized slide rules have the advantage of not requiring any electrical power or the like for their operation. 14 2 Mechanical analog computers Fig. 2.4. Nuclear Weapon Effects Computer “The object of a planimeter is to measure an area; it has, therefore, to solve a geometrical problem by mechanical means.” Measuring areas enclosed by some “good-natured” boundary curve is an important task in many branches of science as well as in commercial applications, registers of real estate and many more. A typical early application was to analyse pressure/volume indicator diagrams23 as those written by recording steam engine indicators,24 which requires the determination of the area enclosed by a curve, which is either plotted in a Cartesian or more often a polar coordinate system. A simple and direct method for performing this task is to cut out the area to be determined and weigh the resulting piece of paper yielding quite good results. Although this method is sometimes still used by chemistry students who regularly 23 See [Hütte 1926, pp. 380 f.]. 24 The first of these devices was invented by James Watt’s25 assistant John Southern26 around 1796 (cf. [Miller 2011]). 2.4 Planimeters 15 have to determine integrals over curves generated by spectrometers and the like, it is not really suitable for everyday usage. As early as 1814 J. M. Hermann,27 a Bavarian engineer, invented a planimeter that was built, after improvement by Lämmle, in about 1817. Unfortunately, this instrument seems to have gone unnoticed by his contemporaries and had no obvious influence on subsequent developments.28 In 1824 an Italian professor for mathematics, Tito Gonnella,29 invented a wheel-and-cone planimeter that used a friction-wheel integrator (see section 2.5.3) to perform the necessary integration.30 The first planimeter that was put into production was a device developed by a Swiss engineer named Johannes Oppikofer,31 who developed two wheel-and-cone planimeters in 1827 and 1836 and a planimeter based on a friction-wheel rolling on a disk in 1849. In fact, there is a plethora of different planimeter principles and implementation variants. The most successful type of planimeter is the polar planimeter that was developed in 1854 by the Swiss mathematician Jacob Amsler-Laffon.32 Figure 2.5 shows a typical polar planimeter,33 which is of a much simpler construction than most of the other instrument types.34 The basis of operation for planimeters in general is Green’s theorem, which relates a double integral over a closed region, i. e., the area to be determined, to a line integral over the boundary of this region.35 Thus, a planimeter is a mechanization of  I ZZ  ∂Q ∂P − dA = F dr. ∂x ∂y Choosing P and Q in a way that the difference under the left integral equals one yields the area sought. Interestingly, it seems that the first explanation of the operation of planimeters using Green’s formula wasn’t given until [Ascoli 1947].36 27 1785–1841 28 Cf. [Henrici 1894, p. 505]. 29 1794–1867 30 [Haeberlin et al.2011] describes this instrument. See also [Henrici 1894, p. 500]. 31 09/15/1782–04/21/1864 32 11/11/1823–01/03/1912 33 [Foote et al. 2007] shows how to build a simple polar planimeter. 34 One notab