• Technologie

Calculating without electricity

Calculating machines as forerunners of the computer

14. Juni 2018

Just imagine: there’s a power cut, your monitor goes black and the lights go off. There you are, sitting in the dark. At ETH Zurich, a computer program that has been performing calculations for two days crashes. – That would not have happened to scientists in the early centuries as they calculated using their heads and a pen and paper. – Or did they? How did they manage to make ground-breaking discoveries in mathematics, astronomy or physics without any electricity?

So-called arithmetic books contain early instructions on calculating, a few copies of which dating from the 16th century are in ETH Library’s holdings. Although these publications were chiefly aimed at traders, they were also used in surveying engineering, astronomy and schools. It was important to the authors to pen their books in German in a practically relevant way and enrich them with a wealth of examples. They regarded mathematics as a foundation for other arts. In the preface to his book, Adam Ries refers to Plato, whose answer to the question as to what sets man apart from the animals was that he is able to calculate and understands numbers. 1

Besides theoretical explanations, concrete calculation aids were also important. ETH Library owns calculating machines that work without electricity – as part of the Collection of Astronomical Instruments. Based on selected devices, this story traces snippets of their development. In doing so, it reveals that calculating machines were forerunners of the computer thanks to their continual improvement and expansion.

Tools for mathematical wizards

Even in school, pupils already enjoyed using abacuses and slide rules centuries ago. 2 Presumably, however, many would have given anything to have devices at their disposal that relieved them of all that heavy thinking…

For thousands of years, people have been inventing tools to help tackle more complex tasks and calculate more swiftly in science, commerce 3 and other walks of life. The quest for straightforward controls and rapid calculation was always at the heart of developing calculation aids: from counting with fingers to today’s computers. However, certain historical calculation aids also have their uses in this day and age, as ETH Zurich Professor Roy Wagner explains.

Calculation aids – a business model

Before long, calculation aids were manufactured for various professions and purposes: as well as conventional logarithmic slide rules, for example, there were also slide rules to perform tachymetric calculations or especially for mechanical and electrical engineers. 4 Just as software is developed by specialist companies for different applications and professions today, calculating devices were already tailor-made for various functions and customer groups in the 19th century. This went hand in hand with advertisements in magazines and books.

The world's most efficient calculating machine „MILLIONAER“The calculator MILLIONAER was devised by engineer Hans Walter Egli (1862-1925), who not only launched the machine on the market, but also founded a company. Along with Bayerische Hypotheken- und Wechselbank in Munich, the materials management department of the Federal Polytechnic Institute became one of the first customers to buy a MILLIONAER. 5 Schweizerische Bauzeitschrift, 1910, volume 55, issue 1, p. 11
„Addiator“, the unsurpassed practical calculating machineNebelspalter: Das Humor- und Satire-Magazin, 1927, volume 53, issue 3, p. 8
MONROE dual counter adding and balancing machineDu: Kulturelle Monatsschrift, 1953, volume 13, issue 9, p. 1
Galaxy 40X – So your students have even more fun with mathematics.Schweizer Schule, 1991, volume 78, issue 9, p. 38

The ability to operate calculation aids was already a prerequisite for success in everyday professional life: “All of us technical officers over the age of 18 must be adept at calculating with slide rules and using tables; otherwise, one faces immediate dismissal.” 6 This is still the case today, albeit worded differently: you would be hard-pushed to find an industry in which computer skills are not an absolute must.

Calculating machines and calculation aids in general were not only used broadly in industry and management, but also in research. Especially mechanical calculating machines were a particularly popular “tool to tackle time-consuming calculation work” in the 19th and 20th centuries. 7

Calculating devices – precursors to calculating machines

Today, the objects in the Collection for Scientific Research and University Teaching compiled by ETH Zurich’s first professor of astronomy, Rudolf Wolf 8 , are part of ETH Zurich’s academic cultural heritage. Besides a vast range of scientific instruments, the 296 items in the collection from the former Swiss Federal Observatory (Eidgenössische Sternwarte) in Zurich also include a group of calculating devices. 9

Russian counting frameA Russian counting frame, also known as an abacus, serves as an instrument to add and subtract positive numbers. Several horizontal rods are held together by a frame. Beads or pearls are threaded onto the rods, with every single decimal number and every rod representing a decimal power. In the past, abacuses were used at markets to calculate prices and still serve as calculating aids in schools to this day. 10 E-Pics Collection of Astronomical Instruments
Slide ruleThe slide rule 11 is an instrument to perform small numerical calculations mechanically and yields approximate results. These devices were extremely important for technology as the majority of calculations were approximate and did not require any strict mathematical precision. 12 E-Pics Collection of Astronomical Instruments
Napier slide rulesNapier slide rules are important versions of slide rules. They are based on the grid method Gelosia, a multiplication and division method. The multiplication table is attached to all four sides of the wooden rods, whereby J stands for the number 1. In order to carry out calculations, the rods are laid next to each other. 13 E-Pics Collection of Astronomical Instruments
Proportional angleProportional angles are based on theories of intersecting lines and work with proportions. Unlike proportional compasses, they have a fixed pivot point (hinge) and are not adjustable. 14 E-Pics Collection of Astronomical Instruments
24-metre loga calculating cylinderUntil the 1970s the loga calculating cylinder with a scale length of 24 metres was the world’s largest and most accurate calculating instrument. From the second half of the 19th century, it was particularly used in banks, insurance companies, schools, industry, research and the military. 15 The example from ETH Zurich’s Department of Computer Science is over 100 years old. (Image: ETH Library)

Devices become machines – early pioneers as forerunners of computers

Scientists such as Wilhelm Schickard 16 (1592-1635), Blaise Pascal (1623-1662), Gottfried Wilhelm Leibniz 17 (1646-1716) and Caspar Schott (1608-1666) or the insurance specialist Charles Xavier Thomas (1785-1870) already developed devices in early centuries that can be described as precursors of today’s computers. Illustrations of these and other inventors can be found in old books.

The Englishman Charles Babbage developed a forerunner to the computer with his “arithmetic machine” in the early 19th century. 18 He saw significant advantages in business considerations: avoiding mistakes, the ability to work at the same pace for hours on end, rapidity and lesser input from humans:

'Charles Babbage' by RoffeCredit: Wellcome Collection. CC BY

The object which Mr Babbage had in view in constructing this new machinery, was to produce printed copies of any mathematical tables, without the possibility of an error existing a single copy. […] Another time it produced forty-four figures in a minute; and, as the machine may be moved uniformly by a weight, this rate of computation may be maintained for any length of time; and it is probable that few writers are able to copy, with equal speed, for many hours together. […] There is one circumstance in the construction of this machine, which is of considerable importance in making larger ones […], any error produced by accident, or by a slight inaccuracy in one of them, is corrected as soon as it is transmitted to the next, and in such a manner as effectually to prevent any accumulation of small errors from producing a wrong figure in the calculation. 19

— Charles Babbage

He compared it to “the most stupendous monument of arithmetical calculation which the world has yet produced”, which had been performed by Gaspard de Prony on behalf of the French government. Babbage explained that thanks to the machine, only 12 of the 96 people used would actually have been needed.

After this calculating machine, which was known as the difference machine, Babbage described an analytical machine in 1837 that took him a major step closer to the computer, even if the machine could not be built. (Konrad Zuse eventually succeeded in building the first working computer, the Z3, around 100 years later 20 .) Ada Lovelace – a British mathematician and programmer – developed a complex program for Babbage’s “Analytical Engine”. 21 She had realized that the analytical machine worked with algorithms and thus autonomously, and that its usage could go beyond calculating operations. 22

With the onset of industrialisation, the number of inventions and the amount of enthusiasm for machines ballooned. For instance, calculators were also showcased at the first World Fair, “The Great Exhibition of the Works of Industry of All Nations”, in London in 1851. 23 A model by Polish inventor Izrael Abraham Staffel 24 was deemed the best and – after he had already received several awards in previous years – he won a medal for it. Besides adding, subtracting, multiplying and dividing, the machine could also extract square roots. 25

Two calculating machines at ETH Zurich

The term “calculating machine” refers to calculating devices where the tens are transferred automatically. 26 ETH Zurich’s Collection of Astronomical Instruments contains two rare calculating machines: a key-driven adding machine by Jean-Baptiste Schwilgué and an arithmometre by Charles Xavier Thomas de Colmar. 27

On 18 November 1820, Thomas de Colmar, an inventor and insurance director in Paris, was granted a patent for a “machine appelée arithmomètre, propre à suppléer à la mémoire et à l’intelligence dans toutes les opérations d’arithmétique”. 28 He also attended the 1851 World Fair in London with his refined arithmometre and received a medal: the judging panel rated it as the second best behind Staffel’s. 29 The machine developed in 1822 was the first calculating machine to be produced in larger batches and launched on the market for a wider pool of consumers. 30 Nevertheless, every single piece can be regarded as a unique specimen as the devices were constantly improved during serial production. 31 The importance of this machine primarily lies in its role-model character as it constituted the prototype for all subsequent advanced addition machines 32 and was thus a major milestone in the history of calculation technology. 33

Unlike the Thomas Arithmometre, the Schwilgué key-driven adding machine" is only capable of addition. Along with an older example in the Historical Museum in Strasbourg, the machine from 1851 at ETH Zurich is the oldest surviving key-driven calculating machine in the world. 34 It was built by Jean-Baptiste Schwilgué (1776-1856), a clockmaker and mathematics teacher from Strasbourg, who filed a patent application for his adding machine, the additionneur mécanique, in 1844 35 . This was one of the first key-driven calculators on the long road towards today’s ubiquitous keyboard 36 .

Schwilgué’s key-driven adding machine was used in everyday accounting work. Like the later common key adder, Bruderer considers ETH Zurich’s copy ideal for accounting and especially totting up long number sequences, known as columns. After all, more powerful machines, such as the Thomas Arithmometre, were too cumbersome and expensive for this purpose. 37

There is no record of these two machines being used in the work of the Swiss Federal Observatory (Eidgenössische Sternwarte) and the Institute of Astronomy. It was only when their technical limitations made them difficult to use in practice that the two machines became an official part of the Collection of Astronomical Instruments: the Schwilgué key-driven adding machine through the undated handwritten addendum to page 187 of the index, the Thomas Arithmometre as recently as 1980, when the Eidgenössische Sternwarte was closed down.

Scientific calculation at ETH Zurich

Many institutes at ETH Zurich had calculation aids that worked without electricity and were used in everyday academic life until the 1970s. This especially affected calculations, be they in meteorology, nuclear physics or astronomy. The calculating machines – and especially their refinement – were thought to harbour great potential for science. 38 The mathematician John von Neumann (1903-1957) was one of the first to hold the belief that “calculations made with the aid of swift electronic calculating machines would make it considerably easier to solve difficult, unsolved problems […]”. 39 Sure enough, electrically powered calculators accelerated scientific calculations and projects from the mid-20th century and enabled major progress to be made.

Pioneering work was also achieved at ETH Zurich in this field during the 20th century. The goal was to meet the growing research needs. 40 With Konrad Zuse’s 41 Zuse Z4”, ETH Zurich became the first university in continental Europe to start using an electronic calculating machine in 1950. 42 ETH Zurich’s first home-grown developments in computer-based calculation, however, stemmed from the Department of Mathematics, where an electronic vacuum-tube computer called ERMETH (ETH’s electronic calculating machine) was developed at the Institute for Applied Mathematics in the 1950s. Although it had not quite been completed, ERMETH was unveiled at ETH Zurich’s 100th anniversary in 1955.

Relay computer Zuse Z4, first computer at a university in continental EuropePhotography, 1950–1955 (ETH Library Zurich, Image Archive, Ans_00590)
Index and plug-in modules Zuse Z4, Institute for Applied MathematicsPhotography, 1950–1955 (ETH Library Zurich, Image Archive, Ans_03694)
ERMETH calculating machine, Institute for Applied MathematicsPhotography, approx. 1955 (ETH Library Zurich, Image Archive, Ans_00290)
Swissair: Installation of the IBM calculating machinesThe tabulating machine 407 has 120 writing wheels, 116 counter digits and 64 storage positions and is able to read and write 9,000 punch cards within the hour. It can add, subtract and compare as well as press the established total onto special punch cards. Its task is being pinned with cables on a control panel by the programmer; put into the machine, this panel controls the automatic course of the required working process. (Original text in: Swissair Journal March 1958)Photography, 1957 (ETH Library Zurich, Image Archive, Foundation Luftbild Schweiz / photographer: Swissair, LBS_SR03-07883)

As Professor Jörg Waldvogel, a professor emeritus from the Institute for Applied Mathematics (Department of Mathematics) who witnessed the events at the time, recalls:

In my diploma thesis at ETH Zurich’s Institute for Applied Mathematics under Professor Eduard Stiefel in 1962, I calculated flight paths from the earth to the moon on ERMETH. The technique was based on the numerical integration of differential equation systems and could no longer be accomplished with a pencil and paper, even with the aid of mechanical MADAS calculators. It took several hours to calculate one path on ERMETH, which I could only get at night. I’d sleep on an airbed on a table in the machine room during a run. Every night at around 12.30, the alarm sounded and the calculation would grind to a halt: every time the electrical network for Zurich’s tram system was shut down, it caused a voltage swing, which stopped the calculation on ERMETH. Fortunately, I could always restart the program from the same point without losing anything.

— Jörg Waldvogel

For Waldvogel, it is clear that scientific breakthroughs such as the American moon landing in 1969 would not have been possible without the major advancements on electronic computer technology in the 1960s. This is still the case today.

Computing today – supercomputers and gigantic computer capacities

The quantities of data that need to be processed in current research require immense computing capacities compared to that of ERMETH, for example. The fact that we operate in completely different dimensions these days is exemplified by the Swiss National Computing Centre CSCS operated by ETH Zurich in Lugano 43 , which is home to Piz Daint, Europe’s most powerful supercomputer 44 . The European Commission has issued a declaration of intent on high performance computing (HPC) 45 , which will serve as a foundation to build a EuroHPC infrastructure in Europe by 2023. 46

For centuries, improving efficiency and reliability was the motivation for developing calculating aids. The data quantities to be processed in industry and academia will continue to rise. As in previous centuries and millennia, people will keep inventing tools that display the necessary capacities and requirements. As a university of technology, ETH Zurich will continue to do its bit. Whether these aids will be added to one of ETH Library’s collections for posterity remains to be seen.


  • Abacus: Simple mechanical calculating tool consisting of a frame with parallel rods along which beads with a hole drilled through them can be pushed back and forth. 47 back >
  • Analytical machine: Design for a mechanical calculating machine for general applications. By British mathematics professor Charles Babbage (1791–1871). Constitutes a major milestone in the history of the computer. 48 back >
  • Arithmetic machine: Machine that works by operating natural numbers. 49 back >
  • Decimal carry: This concerns the intermediate step necessary when counting in positional notation systems. As soon as the number of elements in a position reaches the base number of the positional notation system, they are removed and replaced with a one in the next highest position; in other words, an aggregation or deletion takes place with the carry-over. 50 back >
  • High Performance Computing (HPC): High performance computer which encompasses all computational work where the processing requires high computational power or storage capacity. 51 back >
  • Slide disc: Like slide rules but round. 52 back >
  • Slide rule: Analogue calculating aid to conduct basic arithmetic calculations mechanically and graphically, preferably multiplication and division. 53 back >
  • Tachymetric: Stems from tachymetry and is a form of “quick measurement”. It enables the position and height to be recorded simultaneously. 54 back >
  • Vacuum-tube computer: First-generation computer. Central circuit components consisting of electron tubes. 55 back >
  • Zuse Z4: First commercial computer. 56 back >

  1. Ries, Adam; Helm, Erhard: Rechenbuch, uff Linien unnd Ziphren, Frankfurt: bei Chr.[istian] Egen.[olff] Erben, 1565, p. 7. ↩︎
  2. It was not until the end of the 18th century that the algorists, who calculated using Hindu-Arabic numerals, managed to prevail against the abacists: http://www.library.ethz.ch/en/ms/Virtuelle-Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Bedeutung-Fibonaccis-fuer-die-Gegenwart/Streit-zwischen-Abakisten-und-Algoristen [viewed on 05.03.2018]. ↩︎
  3. Traders use a counting frame, the abacus: http://www.library.ethz.ch/en/ms/Virtuelle-Ausstellungen/Fibonacci.-Un-ponte-sul-Mediterraneo/Das-indisch-arabische-Zahlensystem [viewed on 05.03.2018]. ↩︎
  4. Mayer, Joh. Eugen: Das Rechnen in der Technik und seine Hilfsmittel: Rechenschieber, Rechentafeln, Rechenmaschinen usw. (Vol. 405, Sammlung Göschen), Leipzig: Göschen, 1908, pp. 38-39. ↩︎
  5. After a sluggish start, the demand ballooned to such an extent at the end of the century that Egli returned to Zurich from Munich and opened a workshop. By 1904, Egli was employing 80 people at Albisstrasse 2 in Zürich-Wollishofen. Thanks to the “advanced and open mindset” of the Americans, sales soon took off in the USA, too, where the machines sometimes ran “24 hours a day, seven days a week”. In 1911, Egli launched the first “motorised” MILLIONAER on the market as electrification was increasingly catching on. Continual refinement, the consideration of new customer needs, but also the launch of new machines were the cornerstones of Egli’s huge success. In the early 20th century, it was already recognised that the people who operated machines had to be relieved of as much brain-work as possible. “The more calculating machines spread, the more urgent it became to eliminate any manipulations that required even the slightest mental exertion from the person calculating. (All the information stems from the biographical dossier on Hans Walter Egli in the ETH Zurich University Archives. They contain a copy of the draft for an anniversary publication entitled 50 Jahre EGLI-Rechenmaschinen (“50 Years of Egli’s Mechanical Calculators”) from 1942/43). ↩︎
  6. This sentence stems from a letter which the director of a machine factory wrote to engineer and author Häder. In: Mayer, 1908, p. 6. ↩︎
  7. Graef, Martin: 350 Jahre Rechenmaschinen, München: Hanser, 1973, p. 7. ↩︎
  8. Not only did Rudolf Wolf found the Polytechnic Institute’s observatory; he was also the first Director of ETH Library. Many works in its old holdings stem from his collecting activities. The collection Rudolf Wolf’s Personal Library is available on the platform e-rara.ch. ↩︎
  9. Information on the digitization of the collection is available on the blog ETHeritage. ↩︎
  10. http://modellsammlung.uni-goettingen.de/index.php?lang=de&r=11&sr=51&m=900 [viewed on 15 May 2018]. And: Bruderer, Herbert: Meilensteine der Rechentechnik: Zur Geschichte der Mathematik und der Informatik. Berlin: De Gruyter Oldenbourg, 2015, p. 77. ↩︎
  11. A slide rule from around 1860 is listed in the Verzeichniss der Sammlungen der Zürcher-Sternwarteunder No. 191 as “[…] a copy of a curious slide rule which I obtained from the Horner personal papers and which I commissioned Mr Kern in Aarau to make. […]”.. In: Wolf, Rudolf: Verzeichniss der Sammlungen der Zürcher-Sternwarte, Eidgenössische Sternwarte Zürich, 1873, pp. 76–77. ↩︎
  12. Mayer, 1908, S. 5. ↩︎
  13. Bruderer, 2015, p. 78. ↩︎
  14. Bruderer, 2015, S. 85. ↩︎
  15. Bruderer, 2015, S. 333. ↩︎
  16. cf. Freytag-Löringhoff, Bruno von; Seck, Friedrich: Wilhelm Schickards Tübinger Rechenmaschine von 1623 (5. erw. Aufl., Vol. Heft 4, Kleine Tübinger Schriften), Tübingen, 2002. ↩︎
  17. Walsdorf, Ariane: Die Leibniz-Rechenmaschine der Gottfried Wilhelm Leibniz Bibliothek (Vol. 1, Schatzkammer), Hannover: Gottfried Wilhelm Leibniz Bibliothek, 2014. ↩︎
  18. However, the machine was never built; merely a smaller model. ↩︎
  19. Babbage, Charles: On machinery for calculating and printing, 1822, No. XIV, Vol. VII, pp. 274-281. In: Edinburgh Philosophical Journal: Exhibiting a View of the Progress of Discovery in Natural Philosophy, Chemistry, Natural History, Practical Mechanics, Geography, Navigation, Statistics, and the Fine and Useful Arts, Edinburgh: Constable, 1819-1826. ↩︎
  20. https://de.wikipedia.org/wiki/Zuse_Z3 [viewed on 27.03.2018]. ↩︎
  21. https://de.wikipedia.org/wiki/Analytical_Engine [viewed on 27.03.2018]. ↩︎
  22. https://de.wikipedia.org/wiki/Ada_Lovelace [viewed on 27.03.2018]. ↩︎
  23. Reports by the juries on the subjects in the thirty classes into which the exhibition was divided: Exhibition of the Works of Industry of All Nations, 1851, London: Clowes, 1852. ↩︎
  24. https://en.wikipedia.org/wiki/Izrael_Abraham_Staffel [viewed on 06.03.2018]. ↩︎
  25. A description of how Staffel’s calculating machine works is provided on p. 310f of the 1851 World Fair’s Report by the juries. Another show involving a vast range of calculating aids took place in London on the occasion of an international exhibition of scientific instruments. In contrast to earlier exhibitions, the one in Kensington Palace was more about exchanging scientific ideas than trade. New and historical instruments – slide rules, disc rules and calculating machines – were showcased to illustrate their development. The importance of the exhibition is also reflected in the fact that the catalogue was translated into German. For instance, it also mentions “Napier’s Bones”, which are part of the Collection of Astronomical Instruments. ↩︎
  26. Mayer, 1908, p. 67. ↩︎
  27. In his 2015 book Meilensteine der Rechentechnik (“Milestones in Computer Technology”), Herbert Bruderer collates the results of the research on the two machines. This also yielded instructions for the machines. The following section is based on these latest findings and the sources he compiled. ↩︎
  28. Bruderer, 2015, p. 322. ↩︎
  29. Reports by the juries on the subjects in the thirty classes into which the exhibition was divided: Exhibition of the Works of Industry of All Nations, 1851, London: Clowes, 1852, p. 440. ↩︎
  30. By 1878 1,500 machines had been sold in various different models; they were manufactured from around 1850 until into the 20th century. Cf. Bruderer, 2015, p. 322. ↩︎
  31. Bruderer, 2015, p. 322. ↩︎
  32. Mayer, 1908, p. 73: “The Thomas Arithmometre, after it was used more generally, i.e. the production of calculating machines had gradually become a branch of industry, was now honed and perfected in its components.” On this basis, for example, the Burkhardt Arithmometre was developed in the early 20th century, one of the best and technically accomplished calculating machines. ↩︎
  33. The intensive source work that technology historian Herbert Bruderer conducted reveals that no operating instructions or any other documents of the Thomas Arithmometre can be found. Although much has been published on the arithmometre, nowadays we are reliant on the scientific studies and findings of specialists in history of technology to understand and operate this machine. ETH Zurich’s Research Collection contains a large number of publications by Herbert Bruderer. ↩︎
  34. Bruderer, 2015, p. 314. ↩︎
  35. Bruderer, 2015, p. 315. ↩︎
  36. It took more than 200 years from the invention of the calculating machine by Wilhelm Schickard (1623) to the recording of numbers by keyboard as it is taken for granted today. Cf. Bruderer, 2015, p. 318. ↩︎
  37. Bruderer, 201 pp. 317-318. ↩︎
  38. Neumann, John von: Die Rechenmaschine und das Gehirn, 1965, München: Oldenburg, p. 9 (foreword by Klara von Neumann). ↩︎
  39. Neumann, 1965, pp. 9-10. ↩︎
  40. Information on the historical developments is available on the websites of the Department of Mathematics and Department of Computer Science. ↩︎
  41. Bruderer, Herbert: Konrad Zuse und die Schweiz, Zürich: Eidgenössische Technische Hochschule Zürich, Departement Informatik, Professur für Informationstechnologie und Ausbildung, 2011. And: Bruderer, Herbert: Konrad Zuse und die Schweiz. Wer hat den Computer erfunden? München: Oldenburg, 2012. ↩︎
  42. https://www.inf.ethz.ch/de/departement/geschichte/informatik-forschung.html [viewed on 22.05.2018]. ↩︎
  43. www.cscs.ch [viewed in 05.03.2018]. ↩︎
  44. https://www.newsd.admin.ch/newsd/message/attachments/51291.pdf [viewed on 05.03.2018]. ↩︎
  45. https://ec.europa.eu/digital-single-market/en/news/eu-ministers-commit-digitising-europe-high-performance-computing-power [viewed on 05.03.2018]. ↩︎
  46. https://ec.europa.eu/digital-single-market/en/news/european-declaration-high-performance-computing [viewed on 05.03.2018]. ↩︎
  47. https://de.wikipedia.org/wiki/Abakus_(Rechenhilfsmittel) [viewed on 22.05.2018] ↩︎
  48. https://de.wikipedia.org/wiki/Analytical_Engine [viewed on 22.05.2018] ↩︎
  49. https://de.wikipedia.org/wiki/Arithmetik [viewed on 22.05.2018] ↩︎
  50. http://www.rechnerlexikon.de/artikel/Zehner%FCbertrag [viewed on 15.05.2018] ↩︎
  51. https://de.wikipedia.org/wiki/Hochleistungsrechnen [viewed on 22.05.2018] ↩︎
  52. https://de.wikipedia.org/wiki/Rechenscheibe [viewed on 22.05.2018] ↩︎
  53. https://de.wikipedia.org/wiki/Rechenschieber [viewed on 22.05.2018] ↩︎
  54. https://de.wikipedia.org/wiki/Tachymetrie [viewed on 22.05.2018] ↩︎
  55. https://de.wikipedia.org/wiki/Röhrencomputer [viewed on 22.05.2018] ↩︎
  56. Bruderer, 2015, S. 383. ↩︎