Slide Rules. Edmund Gunter devised in 1620 a method for calculation that used a single log scale with dividers along a linear scale; this anticipated key elements of the first slide rule described by William Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Oughtred [O32] in 1632. A very large variety of slide machines were later constructed.

^ Pascaline: Pascal’s Wheeled Calculator. Blaise Pascal [P45] developed in 1642 a calculator known as the Pascaline that could Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems calculate all four arithmetic operations (addition, subtraction, multiplication, and division) on up to eight digits. A wide variety of mechanical devices were then developed that used revolving drums or wheels (cogwheels or pinwheels) to Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems do various arithmetical calculations.

Stepped Drum Calculators. Gottfried Wilhelm von Leibniz developed in 1671 an improved calculator known as the Stepped Reckoner, which used a cylinder known as a stepped drum with nine Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems teeth of different lengths that increase in equal amounts around the drum. The stepped drum mechanism allowed use of moving slide for specifying a number to be input to the machine, and Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems мейд use of the revolving drums to do the arithmetic calculations. Charles Xavier Thomas de Colbrar developed in 1820 a widely used arithmetic mechanical calculator based on the stepped drum known as Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems the Arithmometer. Other stepped drum calculating devices included Otto Shweiger’s Millionaire calculator (1893) and Curt Herzstark's Curta (early 1940s).

Pinwheel Calculators. Another class of calculators, independently invented by Frank Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems S. Baldwin and W. T. Odhner in the 1870s, is known as pinwheel calculators; they used a pinwheel for specifying a number input to the machine and use revolving wheels to do the arithmetic Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems calculations. Pinwheel calculators were widely used up to the 1950s, for example in William S. Burroughs’s calculator/printer and the German Brunsviga.

Digital Mechanical Devices for Mathematical Tables and Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Functions

Babbage’s Difference Engine. Charles Babbage [B20,B25] in 1820 invented a mechanical device known as the Difference Engine for calculation of tables of an analytical function (such as the logarithm) that summed the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems change in values of the function when a small difference is мейд in the argument. That difference calculation required for each table entry involved a small number of simple arithmetic computations Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems. The device мейд use of columns of cogwheels to store digits of numerical values. Babbage planned to store 1000 variables, each with 50 digits, where each digit was stored by a unique cogwheel. It used cogwheels in Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems registers for the required arithmetical calculations, and also мейд use of a rod-based control mechanism specialized for control of these arithmetic calculations. The design and operation of the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems mechanisms of the device were described by a symbolic scheme developed by Babbage [B26]. He also conceived of a printing mechanism for the device. In 1801, Joseph-Marie Jacquard invented an automatic loom that мейд use Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems of punched cards for the specification of fabric patterns woven by his loom, and Charles Babbage proposed the use of similar punched cards for providing inputs to his machines. He Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems demonstrated over a number of years certain key portions of the mechanics of the device but never completed a complete function device.

^ Other Difference Engines. In 1832 Ludgate [B32] independently designed, but did not Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems construct, a mechanical computing machine similar but smaller in scale to Babbage’s Analytical Engine. In 1853 Pehr and Edvard Scheutz [L90] constructed in Sweden a cog wheel mechanical calculating device (similar to Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems the Difference Engine originally conceived by Babbage) known as the Tabulating Machine, for computing and printing out tables of mathematical functions. This (and a later construction of Babbage’s Difference Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Engine by Doron Swade [S91] of the London Science Museum) demonstrated the feasibility of Babbage’s Difference Engine.

^ Babbage’s Analytical Engine. Babbage further conceived (but did not attempt to construct) a mechanical Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems computer known as the Analytical Engine to solve more general mathematical problems. Lovelace’s extended description of Babbage’s Analytical Engine [L43] describes, in addition to arithmetic operations, also mechanisms for looping and Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems memory addressing. However, the existing descriptions of Babbage’s Analytical Engine appear to lack the ability to execute a full repertory of logical and/or finite state transition operations required for general computation Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems. Babbage’s background was very strong in analytic mathematics, but he (and the architects of similar cog-wheel based mechanical computing devices at that date) seemed to have lacked knowledge of sequential Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems logic and its Boolean logical basis, required for controlling the sequence of complex computations. This (and his propensity for changing designs prior to the completion of the machine construction) might have been Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems the real reason for the lack of complete development of a universal mechanical digital computing device in the early 1800’s.

^ Subsequent Electromechanical Digital Computing Devices with Cog-wheels. Other electromechanical computing Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems devices (see [ER+50]) that contain cog-wheels, included Howard Aiken's Mark 1 [79] constructed at Harvard University and Konrad Zuse's Z series computer constructed in Germany.

^ Mechanical Devices for Timing, Sequencing and Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Logical Control

We will use the term mechanical automata here to denote mechanical devices that exhibit autonomous control of their movements. These can require sophisticated mechanical mechanisms for timing, sequencing and logical control.

^ Mechanisms Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems used for Timing Control. Mechanical clocks, and other mechanical device for measuring time have a very long history, and include a very wide variety of designs, including the flow of liquids Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems (e.g., water clocks), or sands (e.g., sand clocks), and more conventional pendulum-and-gear based clock mechanisms. A wide variety of mechanical automata and other control devices make use of mechanical Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems timing mechanisms to control the order and duration of events automatically executed (for example, mechanical slot machines dating up to the 1970s мейд use of such mechanical clock mechanisms to control the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems sequence of operations used for payout of winnings). As a consequence, there is an interwoven history in the development of mechanical devices for measuring time and the development of devices Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems for the control of mechanical automata.

^ Logical Control of Computations. A critical step in the history of computing machines was the development in the middle 1800’s of Boolean logic by George Boole Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems [B47, B54]. Boole innovation was to assign values to logical propositions: 1 for true propositions and 0 for false propositions. He introduced the use of Boolean variables which are assigned these values, as well the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems use of Boolean connectives (and and or) for expressing symbolic Booelan logic formulas. Boole's symbolic logic is the basis for the logical control used in modern computers. Shannon [S38] was Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems the first to make use of Boole's symbolic logic to analyze relay circuits (these relays were used to control an analog computer, namely MIts Differential Equalizer).

^ The Jevons’ Logic Piano Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems: A Mechanical Logical Inference Machine. In 1870 William Stanley Jevons (who also significantly contributed to the development of symbolic logic) constructed a mechanical device [J70,J73] for the inference of logical proposition that used Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems a piano keyboard for inputs. This mechanical inference machine is less widely known than it should be, since it may have had impact in the subsequent development of logical control mechanisms for Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems machines.

Mechanical Logical Devices used to Play Games. Mechanical computing devices have also been constructed for executing the logical operations for playing games. For example, in 1975, a group of MIT undergraduates Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems including Danny Hillis and Brian Silverman constructed a computing machine мейд of Tinkertoys that plays a perfect game of tic-tac-toe.

^ Mechanical Devices used in Cryptography

Mechanical Cipher Devices Using Cogwheels. Mechanical Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems computing devices that used cogwheels were also developed for a wide variety of other purposes beyond merely arithmetic. A wide variety of mechanical computing devices were developed for the encryption and decryption of secret messages Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems. Some of these (most notably the family of German electromechanical cipher devices known as Enigma Machines [HS+98] developed in the early 1920s for commercial use and refined in the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems late 1920s and 1930s for military use) мейд use of sets of cogwheels to permute the symbols of text message streams. Similar (but somewhat more advanced) electromechanical cipher devices were used by Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems the USSR up to the 1970s.

Electromechanical Computing Devices used in Breaking Cyphers. In 1934 Marian Rejewski and a team including Alan Turing constructed an electrical/mechanical computing device known as the Bomb, which had an Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems architecture similar to the abstract Turing machine described below, and which was used to decrypt ciphers мейд by the German Enigma cipher device mentioned above.

^ Mechanical and Electro-Optical Devices for Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Integer Factorization

Lehmer’s number sieve computer. In 1926 Derrick Lehmer [L28] constructed a mechanical device called the number sieve computer for various mathematical problems in number theory including factorization of small Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems integers and solution of Diophantine equations. The device мейд use of multiple bicycle chains that rotated at distinct periods to discover solutions (such as integer factors) to these number theoretic problems.

^ Shamir’s TWINKLE. Adi Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Shamir [S99a, S99b,LS00] proposed a design for a optical/electric device known as TWINKLE for factoring integers, with the goal of breaking the RSA public key cryptosystem. This was Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems unique among mechanical computing devices in that it used time durations between optical pulses to encode possible solution values. In particular, LEDs were мейд to flash at certain intervals of time (where Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems each LED is assigned a distinct period and delay) at a very high clock rate so as to execute a sieve-based integer factoring algorithm.

^ Mechanical Computation at the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Micro Scale: MEMS Computing Devices. Mechanical computers can have advantages over electronic computation at certain scales; they are already having widespread use at the microscale. MEMS (Micro-Electro-Mechanical Systems) are manufactured by lithographic Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems etching methods similar in nature to the processes microelectronics are manufactured, and have a similar microscale. A wide variety of MEMS devices [M02] have been constructed for sensors and actuators, including accelerometers used Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems in automobile safety devices and disk readers, and many of these MEMS devices execute mechanical computation do their task. Perhaps the MEMS device most similar in architecture to the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems mechanical calculators described above is the Recodable Locking Device [PD+99] constructed in 1998 at Sandia Labs, which мейд use of microscopic gears that acted as a mechanical lock, and which was intended for mechanically Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems locking strategic weapons.

^ VI. Future Directions

Mechanical Self-Assembly Processes.

Most of the mechanical devices discussed in this chapter have been assumed to be constructed top-down; that is they are designed and Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems then assembled by other mechanisms generally of large scale. However a future direction to consider are bottom-up processes for assembly and control of devices. Self-assembly is a basic bottom-up Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems process found in many natural processes and in particular in all living systems.

^ Domino Tiling Problems. The theoretical basis for self-assembly has its roots in Domino Tiling Problems (also known Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems as Wang tilings) as defined by Wang [Wang61] (Also see the comprehensive text of Grunbaum, et al, [G87]). The input is a finite set of unit size square tiles, each of whose sides Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems are labeled with symbols over a finite alphabet. Additional restrictions may include the initial placement of a subset of these tiles, and the dimensions of the region where tiles must be placed. Assuming Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems an arbitrarily large supply of each tile, the problem is to place the tiles, without rotation (a criterion that cannot apply to physical tiles), to completely fill the given region so Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems that each pair of abutting tiles have identical symbols on their contacting sides.

^ Turing-universal and NP Complete Self-assemblies. Domino tiling problems over an infinite domain with only a Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems constant number of tiles were first proved by [Berger66] to be undecidable. Lewis and Papadimitriou [LP81] showed the problem of tiling a given finite region is NP complete.

^ Theoretical Models of Tiling Self-assembly Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Processes. Domino tiling problems do not presume or require a specific process for tiling. Winfree [W98] proposed kinetic models for self-assembly processes. The sides of the tiles are assumed to have some Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems methodology for selective affinity, which we call pads. Pads function as programmable binding domains, which hold together the tiles. Each pair of pads have specified binding strengths (a real number on the Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems range [0,1] where 0 denotes no binding and 1 denotes perfect binding). The self-assembly process is initiated by a singleton tile (the seed tile) and proceeds by tiles binding together at Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems their pads to form aggregates known as tiling assemblies. The preferential matching of tile pads facilitates the further assembly into tiling assemblies.

^ Pad binding mechanisms. These provide a mechanism for the preferential matching of tile Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems sides can be provided by various methods:

magnetic attraction, e.g., pads with magnetic orientations (these can be constructed by curing ferrite materials (e.g., PDMS polymer/ferrite composites Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems) in the presence of strong magnet fields)the and also pads with patterned strips of magnetic orientations,

capillary force, using hydrophobic/hydrophilic (capillary) effects at surface boundaries that generate lateral forces,

shape matching (also known Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems as shape complementarity or conformational affinity), using the shape of the tile sides to hold them together.

(Also see the sections below discussion of the used of molecular affinity for pad Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems binding.)

Materials for Tiles. There are a variety of distinct materials for tiles, at a variety of scales: Whitesides (see [XW98] and http://www-chem.harvard.edu/GeorgeWhitesides.html) has Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems developed and tested multiple technologies for meso-scale self-assembly, using capillary forces, shape complementarity, and magnetic forces). Rothemund [R00] gave some of the most complex known meso-scale tiling assemblies using polymer Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems tiles on fluid boundaries with pads that use hydrophobic/hydrophilic forces. A materials science group at the U. of Wisconsin (http://mrsec.wisc.edu/edetc/selfassembly) has also tested meso-scale self-assembly Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems using magnetic tiles

^ Meso-Scale Tile Assemblies. Meso-Scale Tiling Assemblies have tiles of size a few millimeters up to a few centimeters. They have been experimentally demonstrated by a number of methods, such Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems as placement of tiles on a liquid surface interface (e.g., at the interface between two liquids of distinct density or on the surface of an air/liquid interface Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems), and using mechanical agitation with shakers to provide a heat source for the assembly kinetics (that is, a temperature setting is мейд by fixing the rate and intensity of shaker agitation).

Applications of Meso Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems-scale Assemblies. The are a number of applications, including:

Simulation of the thermodynamics and kinetics of molecular-scale self-assemblies.

For placement of a variety of microelectronics and MEMS parts Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems.

Mechanical Computation at the Molecular Scale: DNA Computing Devices. Due to the difficulty of constructing electrical circuits at the molecular scale, alternative methods for computation, and in particular mechanical methods, may provide unique opportunities Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems for computing at the molecular scale. In particular the bottom-up self-assembly processes described above have unique applications at the molecular scale.

^ Self-assembled DNA nanostructures. Molecular-scale structures known as DNA Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems nanostructures (see surveys by Seeman [S04] and Reif [RS07]) can be мейд to self-assemble from individual synthetic strands of DNA. When added to a test tube with the appropriate buffer Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems solution, and the test tube is cooled, the strands self-assemble into DNA nanostructures. This self-assembly of DNA nanostrucures can be viewed as a mechanical process, and in fact can be used Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems to do computation. The first known example of a computation by using DNA was by Adleman [A94,A98] in 1994; he used the self-assembly of DNA strands to solve a small Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems instance of a combinatorial optimization problem known as the Hamiltonian path problem.

^ DNA tiling assemblies. The Wang tiling [W63] paradigm for self-assembly was the basis for scalable and programmable Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems approach proposed by Winfree et al [WL+98] for doing molecular computation using DNA. First a number of distinct DNA nanostructures known as DNA tiles are self-assembled. End portions of the tiles, known Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems as pads, are designed to allow the tiles to bind together a programmable manner similar to Wang tiling, but in this case uses the molecular affinity for pad binding due to hydrogen Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems bonding of complementary DNA bases. This programmable control of the binding together of DNA tiles provides a capability for doing computation at the molecular scale. When the temperature of the test tube containing these Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems tiles is further lowered, the DNA tiles bind together to form complex patterned tiling lattices that correspond to computations.

^ Assembling Patterned DNA tiling assemblies. Programmed patterning at the molecular scale can Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems be produced by the use of strands of DNA that encode the patterns; this was first done by Yan, et al [YL03] in the form of bar-cord striped patterns, and more Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems recently Rothemund [R06] who self-assembled complex 2D molecular patterns and shapes. Another method for molecular patterning of DNA tiles is via computation done during the assembly.

^ Computational DNA tiling Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems assemblies. The first experimental demonstration of computation via the self-assembly of DNA tiles was in 2000 by Mao et al [M00], which provided a 1 dimensional computation of a binary-carry computation (known as prefix Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems-sum) associated with binary adders. Rothemund et al [R04] in 2004 demonstrated a 2 dimensional computational assemblies of tiles displaying a pattern known as the Sierpinski triangle, which is the modulo 2 version of Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Pascal’s triangle.

Other autonomous DNA devices. DNA nanostructures can also be мейд to make sequences of movement, and a demonstration of an autonomous moving DNA robotic device, that moved without outside mediation across Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems a DNA nanostructures was given by Yin et al [Y04]. The design of an autonomous DNA device that moves under programmed control is described in [RS07]. Surveys of DNA Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems autonomous devices are given in [RL09] and [BY07].

Acknowledgements

We sincerely thank Charles Bennett for his numerous suggests and very important improvements to this survey.

^ IX. Bibliography

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[TS+02] Toshinori Munakata, Sudeshna Sinha, and William L. Ditto, Chaos Computing: Implementation of Fundamental Logical Gates by Chaotic Elements, IEEE Transactions on Circuits adn Systems-I Fundamental Theory Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems and Applications Vol. 49, No. 11, pp 1629-1633 (Nov 2002).

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[H50] Douglas R. Hartree, Calculating Instruments and Machines, Cambridge University Press, London, UK (1950).

[ER+50] Engineering Research Associates Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems Staff, High-Speed Computing Devices, McGraw-Hill Book Co. New York City, NY 1950.

[C80] George C. Chase, "History of Mechanical Computing Machinery." IEEE Annals of the History of Computing, Volume 2, No. 3, pp. 198-226, July 1980.

[M Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems92] Ernst Martin, "The Calculating Machines." The MIT Press, Cambridge, Massachusetts, 1992.

[D00] Martin Davis, The Universal Computer: The Road from Leibniz to Turing, Norton Press, Norton, VA, 2000.

[N02] Jeremy M. Norman (Editor Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems), The Origins of Cyberspace: From Gutenberg to the Internet: a sourcebook on the history of information technology, Norman Publishing, Novato, CA, 2002.

[H14] E. M. Horsburgh, Modern Instruments of Calculation, G. Bell & Sons, London Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems (1914), p. 223.

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[S54] Walter A Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems. Soroka, Analog Methods in Computation and Simulation, McGraw-Hill Co., New York, NY (1954).

[FB+06] T. Freeth, Y. Bitsakis X. Moussas, J. H. Seiradakis, A. Tselikas, H. Mangou, M. Zafeiropoulou, R. Hadland, D Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M. G. Edmunds, Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism Slide Rules - Iii. Computational Complexity of Mechanical Devices and their Movement Problems, Nature, Nature 444, 587-591 (30 November 2006).

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[BN47] L. Bragg and J.F.Nye, , “A dynamical model of a crystal structure,”Proc. R. Soc. A,,

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