**Calculating Machines**

The other aspect of this conceptual/technological history is the process of making the calculations by which data might be generated and manipulated for statistical, scientific and eventually for musical and graphical purposes leading to the developments in the arts that I am discussing in this project.

Calculating machines go back as far as the abacus, long used with great facility in China and Japan [**Fig.1]**. In Europe the first aids to calculation were Napier’s Rods^{1}, a “pocket calculator” based on the multiplication tables set up in columns, followed by the slide rule which used logarithms, invented by Napier in 1614.^{2} Of mechanical devices Shickard’s “Calculating Clock”(1623) is said to be the first, followed by Pascal’s eight-digit “Pascaline” (1642) and Leibniz’ devised a multiplier that used a special stepped drum mechanism that he had constructed for him by a French clockmaker in 1674.^{3} Apart from commercial activity, for which most of these machines were intended, navigation was the main requirement for mathematics in the 17th and 18th centuries. Mostly navigation at sea was done with the aid of tables which had to be calculated and reproduced. This process was, of course, constantly prone to both arithmetic and clerical errors and so the tables were known to be unreliable. It was Babbage who, about 1820, first realised that a set of tables might be both mechanically calculated and mechanically printed, thus significantly reducing the opportunity for error and naval disaster. He formulated his “method of differences” and began to build a model of his difference Engine which he completed in 1822.

**Mechanical calculators**

A variety of calculating machines were developed in the 19th century starting with the Thomas de Colmar Arithmometer in 1820 and which remained in production until the end of the 19th century. New types of machines including the Millionaire which added a multiplier to the Arithmomenter, the Odhner and the Brunsviga [**Fig.2**], and the Comptometer, a key operated calculator which greatly sped up operations.^{4} These kinds of machines remained in use until the first world war. After the war calculating and, now, accounting machines built by IBM and the British Tabulating Machine Co., in which data was entered via punched-cards based on the Hollerith card took over much of the work although scientific calculation was still being done on keyboard calculators such as the Burroughs. The use of mechanical calculating machines slowly increased over the decades from the late 19th century until WWII, though they were slow and required large numbers of operators (known as “computers” and mostly women). But many scientific and engineering problems were essentially intractable to these mechanical numerical functions and needed to be solved by geometric or “graphical” means. This class of problems involve calculations on continuously changing signals which reflect the progress of some physical system over time and require the solution of differential equations. They had to be done using differentiators and integrators and awaited the development of a new kind of device; the analogue computer.

**The Harmonic Synthesiser**

In the period from the end of the Second World War until the ubiquity of desk-top computers in the 1980s the machines that were used to solve problems in this graphical/geometrical manner were known as analogue computers. In the late 19th and early 20th centuries, however, these machines were preceded by various versions of the planimeter and the resolver, harmonic analysers and differential analysers. Resolvers used the properties of triangles to measure distances (heights) that could not actually be traversed (like the height and range of an aeroplane in anti-aircraft gunnery).^{5} The planimeter is a means for integrating the motions of two variables^{6} and led to the Tide Predictor and later the Harmonic Synthesizer built by Lord Kelvin (William Thomson) in the 1870’s.^{7} Kelvin’s Tide Predictor was built to predict tide heights by integrating the relative motions of the sun and moon around the spinning earth and biasing these according to the recorded tide heights and timings for a particular segment of the continental shelf. This device is mathematically based in the Fourier transform in which any complex record of change over time can be broken down into a collection of sine-waves whose periods would be determined by the cycle time of the process being explored. Kelvin also discovered that by building mechanical analogue representations of these harmonic functions using disk-and-ball integrators^{8} a system for solving differential equations could be built.^{9} This, although it was not realised as a functional machine due to the very low torque of the output of the disk and ball integrator, led the American engineer, Vannevar Bush, to develop machines “to aid the calculations with continuous functions required by design engineers.”^{10}

**The Differential Analyser**

By employing rotating shafts, leadscrews^{11} and rotating disk integrators (a modification of Kelvin’s disk-and-ball integrators) assisted by the necessary torque amplifiers,^{12} Vannevar Bush developed his Differential Analyser (in the 1930’s).^{13} Douglas Hartree, who went on to be instrumental in the development of digital computers in Britain, built a version of Bush’s Differential Analyser at Manchester University. [**Fig.3**]

A differential analyser consists of a number of units, each of which carries out an operation which can be regarded as a translation into mechanical terms of a process (integration, addition, etc.) which may be required in the mechanical integration of a differential equation, and some means of interconnecting these units. Each unit is driven by the rotation of one or more shafts, and the result of its operation is the rotation of a shaft driven by it. Each shaft represents one of the quantities occurring in the equation to be solved, and the total rotation of each shaft measures the corresponding quantity, on an assigned scale. The interconnections between the units are made in such a way that the relations between the rotations of the various shafts form a translation into mechanical terms of the equation to be solved. Then the rotation of one shaft, representing the independent variable, drives the remainder of the shafts in accordance with the equation represented by these interconnections.”^{14}

Operationally the Differential Analyser consists in a set of Input and Output tables which are linked to disk-and-wheel integrators via a set of buss bars running along a long table. The outputs of the integrators are then linked to further integrators and to output leadscrews via torque amplifiers and the buss, allowing “the input of any mechanism to be connected to the output of any other, as required by the problem being solved.”^{15} Input data as continuous curves is traced on an input table using a stylus connected in two dimensions (*x* and *y*) to leadscrews and the graphical solutions to problems are drawn (*ie*. plotted) by the motions of the leadscrews on the output plotting table (or plotter). The Differential Analyser was capable of solving most second-order differential equations of practical importance to an accuracy of 1-2%.”^{16}

At least two Australian differential analysers were built. The first was David Myers’ mechanical Integraph, **Fig.4** built as his doctoral project in 1938.^{17} Somewhat later, when Myers had become head of the CSIRO Division of Mathematical Instruments, he and Ross Blunden constructed an electro-mechanical differential analyser [**Fig.5**] in the CSIRO Section of Mathematical Instruments in 1950-51.^{18} In a paper describing the CSIRO Differential Analyser they note that with plotting tables the input & output tables are interconvertible, changing their function “involves only the substitution of a pen and the connection of a handwheel”. When used as an output table both leadscrews are driven by elements in the instrument whose rotations are measures of the variable concerned” *ie*, the abscissa and the ordinate of the function.^{19} The idea of the inter-convertibility of the input and output device becomes important when we look at the work of Iain Macleod and Chris Ellyard at the ANU in the early 1970s.

**Gunnery computers**

The Differential Analysers were very expensive since their accuracy depended on very high precision mechanical engineering. However, various types of mechanical problem solving machines were of great importance during WWII, particularly in the calculations related to “the solution of the differential equations of shell trajectories in the preparation of ballistic firing tables”^{20} for aiming artillery at moving targets. Locating an aircraft, determining its range and trajectory and then adjusting for the time it takes to get a shell up to where the aircraft will be requires compensating the gun’s aim for the future (30 seconds or so) position of the aircraft as well as other problems such as wind drift. And all this had to be done in real-time. This requires a servo-mechanism function using the feedback of any errors in the current state to control the predicted positions of future states of the system, in a process of constant updating of the aim. Norbert Wiener’s work on these kinds of problems led to his development of the concept of feedback control systems and Cybernetics and to his 1948 book *Cybernetics or Control and Communication in the Animal and the Machine*.^{21}

Wiener recognised that the feedback principle could cover much of what is involved in various aspects of an animal’s behaviour in its world and our behaviour in the human social world. He showed that this cybernetic principle could be seen to be operating almost ubiquitously in biological and social as well as engineering processes (let alone the use of artillery). The argument here is that this cybernetic principle functions in the many levels of interaction between artists, scientists and engineers and their technologies and, as is explored here, in the development of new computing and imaging technologies through the workings of social processes which can be described effectively within the framework of Deleuzian desiring-machines. Very importantly for the development of artists’ use of many of the technologies that had been bent to or developed for military use in the United States, large numbers of gunnery computers were built and, after the war, appeared on the scrap market where, for example, the Whitney brothers acquired several and built their animation machines from them.^{22} They used them to build controllers that would move lights around in mostly cyclical visual patterns for their earliest (analogue)-computer animated films.

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FOOTNOTES

1 Often known as “Bones” on account of their commonly being made with ivory. Described in Bryden, D.J. (1992) *Napier’s Bones: A History and Instruction Manual*, London: Harriet Wynter Ltd;

2 These devices may be explored further in Williams, Michael R. (1990) “Early Calculation”, chapter 1 of Aspray, 1990 pp.14-20; and Tweedale, Geoffrey (1990) *Calculating Machines and Computers*, Princes Risborough, Bucks, UK: Shire Publications, pp.4-6.

3 Each of which are described in Tweedale, 1990 (*op cit* note 101), pp.7-9; and in considerably more detail in Williams, 1990 (*op cit* note 101), pp.35-49.

4 For more on these devices see Tweedale, 1990 (*op cit* note 101), pp.10-16; and Williams, 1990 (*op cit* note 101), pp.50-57.

5 Bromley, 1990b, *op cit*, pp.156*ff*.

6 Bromley, 1990b, *op cit*, pp.166-71.

7 Bromley, 1990b, *op cit*, pp.172-7.

8 A reasonably accessible description of the integrator mechanism is available in Hartree, 1950, pp.5-6

9 Bromley, 1990b, *op cit*, pp.174-77.

10 Bromley, 1990b, *op cit*, p.179.

11 A shaft having a screw-thread along its length that, by its rotation, can lead a “rider” of some sort back and forth along that length. The rider could be a pen carriage on a mechanical plotter driven by a pair of perpendicular lead screws.

12 Which amplify the power output from a disk-and-wheel integrator up to the power required to drive a load (more shafts and integrators).

13 Bromley, 1990, *op cit*, pp.179-185; and Hartree,1950, *op cit*, p.8 and pp.14-20.

14 Hartree, 1950, *op cit*, p.8.

15 Bromley, 1990, *op cit*, p.181.

16 Bromley, 1990, *op cit*, p.180.

17 The instrument is described in Myers, D.M. (1939). “An Integraph for the Solution of Differential Equations of Second Order”, *Journal of Scientific Instruments*, vol.16, no.7, p.209.

18 Pearcey, Trevor (1988b), *A History of Australian Computing*, Melbourne: Chisholm Institute of Technology, pp.24-6.

19 Myers, D.M. and Blunden, W.R. (1951) “The C.S.I.R.O. Differential Analyser” *Proceedings of a Conference on Automatic Computing Machines*, Sydney: CSIRO Division of Radiophysics, March 1951. p.22. See p.28 Fig.2 for a demonstration of the plotting of a non-linear function.

20 Bromley, 1990, *op cit*, p.190.

21 Wiener, Norbert (1948) *Cybernetics. Or Control and Communications in the Animal and the Machine*. New York, Wiley, 1948.

22 Although this scrap market hardly existed in Australia, it was very important in the development of a number of artists working in art and technology in the U.S. Originally the Whitney brothers and later the Vasulkas took great advantage of the technologies that only became available there for their artistic work.

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