Difference machine

Difference machine no. 1 by Charles Babbage

A difference engine (English difference engine ) is an arithmetic logic unit, with the polynomial functions can be evaluated. The first difference machines were purely mechanical implementations of an algorithm otherwise carried out by humans , which was mainly used in the calculation and expansion of existing tables. Since every continuously differentiable function can be approximated by a polynomial ( approximation ), difference machines can be used in many ways, both for interpolation between table entries and for the recalculation of function values. Despite this generally recognized usefulness, almost 80 years passed from the first machine to the systematic use in table creation.

The first thoughts about such a machine were already in the late 18th century. Independently of this, the idea of ​​mechanizing the creation of tables was picked up in 1812 by Charles Babbage and implemented for the first time in 1820–1822. Babbage's attempts to build a usable machine failed. The publications about his machine, especially that of Dionysius Lardner , however, proved to be very influential and led to numerous machines ( Scheutz , Wiberg , Deacon ), each of which was used for a short time. The eight-digit logarithm tables published by Julius Bauschinger and Johann Theodor Peters in 1910 were the first significant tables created with a difference machine. The difference machine designed by Christel Hamann for the creation of the tables was stolen and the plans were considered to have been lost as early as 1928. Supported by the success of this machine, numerous commercial calculating machines were adapted so that they could also be used as difference machines: In 1912, T. C. Hudson and the Burroughs Adding Machine Company developed a machine for two differences. In 1928 the Brunsviga- Dupla was presented, a calculating machine with two result registers that could be used as a difference machine. In 1931, while inspecting a National Accounting Machine Class 3000 accounting machine , Leslie John Comrie discovered that it could also be used as a six-difference machine . In 1950 Alexander John Thompson built a difference machine to calculate his 20-digit logarithm table by fixing four calculating machines on a wooden support and coupling them mechanically.

With the advent of computers, first the difference machines and later the log tables disappeared.

history

Johann Helfrich von Müller

After completing his calculating machine, Johann Helfrich Müller wrote a user manual in 1786, in the appendix of which he presented an outlook for future improvements. In addition to hints about the usefulness of a printed matter for documenting calculation results, he also presented his thoughts on a new type of machine that should be able to calculate series of numbers using the method of differences. There is no indication that Müller's description was more than a concept. Rather, the description is to be seen as an offer to design and manufacture such a machine including a printer with appropriate funding. He was therefore in correspondence with the mathematician Albrecht Ludwig Friedrich Meister from Göttingen . The first written mention of the basic principles of a difference machine can be found in a letter from Müller to Meister from September 10th, 1784. In recent research it is emphasized that the idea of building such a difference machine originated from Meister: Müller’s statements prove that he had no idea of ​​the possible applications of such a machine.

Charles Babbage

Difference Engine No. 0

The first functional difference machine was built by Charles Babbage between 1820 and 1822. According to today's doctrine, Babbage was not familiar with Johann Helfrich von Müller's considerations at that time, and these were only later translated into English by his friend John Herschel . This first difference machine could calculate with two differences with an accuracy of six digits. Babbage saw the machine as a partial model of a larger differential machine still to be built. The model served as a feasibility study to obtain financial support for a larger machine. The model has been demonstrated many times and its computing speed was stated in a letter to the President of the Royal Society , Sir Humphry Davy , with 44 results per minute. Even if Babbage described the production of the machine as in need of improvement, it should be noted that the machine worked extremely well, mainly due to its conservative design. Even if it was the only calculating machine that Babbage completed during his lifetime, it shows that he was very well able to compensate for the manufacturing problems and inaccuracies of his time with an appropriate design. The machine did not fail to work, a year after its launch, Babbage was given £ 1,500 by the government to develop and manufacture a difference machine .

In the specialist literature, the model is called Difference Machine No. 0 denotes. A detailed description, including functional drawings of machine parts, can be found in a manuscript by Babbage, which Babbage gave to his friend H. W. Buxton for publication. The model, a contemporary representation of the model or its construction plans no longer exist today. It is believed that Henry Prevost Babbage was part of Difference Machine No. 0 for creating the fragments of the difference machine No. 1 (see below) used.

Difference machine No. 1

Demonstrator of the No. Difference Machine, built by Charles Babbage in 1832. 1 ( Science Museum , London).
Cardboard gears made as a functional model by Charles Babbage (1831), (Science Museum, London).

With the guaranteed financial support from the government, Babbage hired precision mechanic Joseph Clement to help him build Difference Machine No. 1 support. Clement was the best metalworker in London at the time. His perfectionism extended to the construction, the drawings and the craftsmanship. With Babbage and Clement, two perfectionists from different fields met each other, but they both agreed that the machine should be of excellent quality rather than fast. Babbage's government contractors had different views, culminating in Babbage and Clement's falling out in 1833 over payment for his services. According to the case law of the time, Babbage was only allowed to keep the manufactured parts, the construction plans of the differential machine and its individual parts as well as the manufactured tools remained with Clement. A special feature of the Difference Engine No. 1 provided a recursion mechanism through which the calculation result, shifted by any powers of ten , could be deducted from the second difference. At this point in time, the parts that had been manufactured by Clement up to then became the demonstrator of the difference machine No. 1 composed. This demonstrator was never developed into a complete difference machine; it can be viewed today in the Science Museum in London. When Clement turned over the engineering drawings to Babbage, his attention was focused on the Analytical Engine , a machine that, if completed, could have completely replaced all difference machines.

. This mechanism could, for. B. Trigonometric functions are calculated and not only existing table values ​​are interpolated. This extension is the first ever recursion mechanism.

Shortly after the demonstrator was built, Dionysius Lardner published an article on Charles Babbage's adding machine in the Edinburgh Review in 1834 . Babbage provided the details for the article, but Lardner decorated and altered them almost at will. The aim of the article was to convince the government of the usefulness of the machine and to guarantee further funding. Even if he missed this goal, he inspired both George Scheutz and Alfred Deacon to design and build their own difference machines.

Fragments from original parts

Even if the difference machine No. 2 and the original parts of Babbage are now the central exhibits in the Science Museum in London in the mathematics and computer department, the museum did not want to include the difference machine, the demonstrators of later difference machines or parts thereof during his lifetime. In 1879, after Babbage's death, his youngest son, Henry Prevost Babbage, put together six separate difference machine fragments from the original parts he found, all of which were manufactured before 1834, and gave them away.

Difference Engine No. 2

In 1849, Babbage returned to his obligations to the government and turned away from the further development of the Analytical Engine to create his Difference Engine No. 2 construct. He wanted to transfer the constructive advances he had made by designing his analytical engine to the difference machines. Babbage was only concerned with making the construction drawings, not actually building the difference machine. He assumed that the government would start building the difference machine No. 2 would carry out independently. By creating the design drawing, he wanted to blame the government for not completing Difference Machine No. 1 settle. The set of construction drawings for the difference machine No. 2 is the only complete set of construction drawings for one of Babbag's machines, all other machines were only incompletely documented. The construction of the difference machine No. 2 had a width and height of 3 meters each and a depth of 1.5 meters. The machine could count on 7 differences with 31 positions each. An attached printer should transfer the calculation results directly to a print matrix .

Construction of the difference machine No. 2 in the Science Museum, London

Only between 1989 and 1991 was in the London Museum Science , the Difference Engine No. 2 built and proven its functionality. In 2000 the printer, also designed by Babbage, was completed. The combination of calculating machine and printer weighs about five tons and was assembled from 8000 bronze and cast parts. In 2008, another Difference Engine No. 2 including printer in the California Computer History Museum and exhibited until January 2016.

It is important to the Science Museum that the two difference machines No. 2 are not replicas, as Charles Babbage never has a functioning whole difference machine No. 2 built. The machines from 1991 and 2008 are therefore referred to as originals.

Georg and Edvard Scheutz

Scheutz prototype from 1843, restored. Wooden frame with moving metal parts. Drive crank front left.
Machine in the Tekniska museet , Stockholm

Scheutz No. 0

In the course of his publishing activities, Scheutz came into contact in 1830 with Charles Babbage's explanations about his difference machine . He was fascinated by the idea of ​​building a machine that could calculate and record the result immediately on printing plates. With the help of a detailed functional description in an overview article by Dionysius Lardner in the Edinburgh Review , Georg Scheutz constructed a model made of wood, wire and cardboard in order to convince himself of the functionality of the principle. In the summer of 1837 he allowed his 16-year-old son, the future engineer Edvard Scheutz (1821–1881), to build a larger model out of metal. Georg Scheutz was so enthusiastic about the possibilities of this model that he presented it to the Royal Swedish Academy of Sciences and asked for financial support for the production of a complete difference machine. The support was not granted.

Edvard Scheutz refined the model further: a difference machine with five digits and one difference was completed in 1840, the expansion to three differences in 1843.

The model from 1843 was sold to the Nordiska Museet , Stockholm for 50 crowns after the death of Edvard Scheutz . The model was rediscovered in the museum's fundus in December 1979 by Michael Lindgren as part of his research for his dissertation and restored as much as possible together with Per Westberg , the museum's furniture restorer. Since the main drive gear already had numerous broken teeth before it was rediscovered and this gear as a central part of the machine should not be replaced as part of the restoration, the machine is currently in a non-functional condition. The model is exhibited in the Tekniska museet , Stockholm.

Scheutz difference machine No. 1. (1853)
machine part of the Smithsonian Institution , USA since 1963

Scheutz No. 1

In 1844, George Scheutz sought financial support from the Swedish Crown to build a complete model of the differential machine . It wasn't until 1851 that he was promised a third of the original amount in case he could demonstrate a fully functioning model. With technical and logistical support from Johan Wilhelm Bergström (1812–1881), Georg and Edvard Scheutz were able to present a functional 15-digit difference machine with a depth of four differences that could print an eight-digit result in October 1853. The machine is also known as Scheutz No. 1 denotes: The Scheutzs' first complete difference machine. Some authors refer to the No. 1 also as the Scheutzs 'second machine, they see the demonstrator from 1843 as the Scheutzs' first difference machine.

In the autumn of 1854, the Scheutzs started on an advertising trip for the difference machine to England. A patent (No. 2216 from 1854) was granted to them on April 13, 1855. The machine was u. a. exhibited, demonstrated and appraised at Somerset House of the Royal Society . The machine was then shown at the Paris World's Fair in 1855 . Charles Babbage showed interest in the Scheutz machine and supported it in both London and Paris in their sales efforts, but not without emphasizing his contribution. He tried in vain to convince the Royal Society (London) to appoint Georg Scheutz as a member.

In 1856, Benjamin A. Gould arranged for the Dudley Observatory in Schenectady , N.Y , to purchase the Difference Machine for £ 1,000 . The machine was delivered in April 1857 and operated for two months the following winter. After these two months, Gould was released from his duties and the machine was no longer used. The Scheutz No. 1 was therefore never used according to its actual purpose, the direct creation of print templates for tables. In 1963 the difference machine was transferred to the Smithsonian Institution .

Scheutz No. 2

Scheutz difference machine No. 2. (1859)
machine in the Science Museum , London.

A second difference machine (Scheutz No. 2), practically a copy of the first machine, was built by Edvard Scheutz on behalf of the British Treasury together with Bryan Donkin in London, England. The machine was handed over on July 5, 1859 after 19 months of construction. The first tables to be printed using the machine were the barometer tables by William Gravatt , 1859. a. the life tables (London, 1864) were calculated using the difference machine. The machine was retired in 1914 and given to the Science Museum in London.

Although the Scheutzs only built two differential machines and these machines did not function properly, they managed to arouse public interest in these two machines. In addition to the exhibitions, the Scheutzs repeatedly demonstrated their machine. In 1857 the Scheutzs printed a 50-page brochure about the possibilities of the difference machine, including a 29-page logarithm table from 1 to 10,000. The book was sent to all possible buyers of a difference machine. A French edition was completed in 1858.

Alfred Deacon

The review by Dionysius Lardner in the Edinburgh Review not only inspired George Scheutz to construct a difference machine, but also, independently of it, Alfred Deacon from London. Its machine could count on three differences and 20 digits. The machine has been lost, but it is possible that it was at least temporarily in the possession of Charles Babbage, who offered a small differential machine from London as an exhibit for the World Exhibition in London (1862) .

Martin Wiberg and George Bernard Grant

The success of George and Edvard Scheutz in the construction of a difference machine led Martin Wiberg (1826–1905) and George Bernard Grant (1849–1917) to their own designs.

Martin Wiberg presented his machine with four differences and an arithmetic unit of 15 digits in 1860. In addition to being honored by the future King of Sweden Oscar II and numerous awards, the machine was presented to the Académie des Sciences at the recommendation of Charles Babbage and was given a favorable assessment. Wiberg's goal was not to sell the machine, but to create scientific tables or their printing templates. Wiberg was only able to publish a logarithm table calculated with the machine in Swedish in 1875 and in German and English in 1876. In the introduction to his panels, Wiberg blamed his efforts on an appealing typography for the delays. The table work was exhibited at the 1876 World's Fair in Philadelphia . The demand was low, the table work is very rare these days. The difference machine is located in the Tekniska museet in Stockholm.

Difference machine by George Bernard Grant, 1876

George Bernard Grant heard about Charles Babbage's difference machines for the first time as a student in 1870. At this point in time he had already made numerous attempts to mechanize computing processes. He completed his demonstrator of a difference machine in 1871 and described it in a publication. After graduating, he continued to work on the difference machine. The aim was to exhibit a working machine for the world exhibition in Philadelphia (1876). The machine was finished a few days beforehand, but it was not fully operational. The machine was 2.5 m long and 1.5 m high and could be operated with a hand crank. If the hand crank was replaced by a belt drive , the computing power doubled from 12 terms per minute to 24. As a concession to its financiers, the machine was transferred to the University of Pennsylvania . The machine went down. A difference machine based on Grant's plans was sold to Provident Mutual Life Insurance Company , where it was used to calculate mortality tables.

Christel Hamann

Difference machine from Christel Hamann

When Julius Bauschinger and Johann Theodor Peters (1889–1941) began a project to create eight-digit instead of the previous seven-digit logarithm tables for the natural numbers and the trigonometric functions , they asked Christel Hamann (1870–1870) to construct a difference machine. 1948). Hamann delivered the machine in 1909. A detailed description of the machine can be found in the foreword to the first volume of the tables. It was a 16-digit machine with two differences and a built-in paper printer. The difference machine was built much more simply than the previous designs: Not only did it work with only two differences, it also had no type of automatic system, i.e. H. the user first had to add the second difference to the first by operating one crank, in order to then add the first difference to the function value with the second crank. Despite this limitation, an experienced user could compute 36 table entries in five minutes. Bauschinger and Peters planned their tables around a difference machine with only two differences, using the machine only for interpolation over small intervals. The machine itself went under; an image can be found on the cover of the panels published in 1910.

Input box, Burroughs machine of the Nautical Almanac Office

Around 1912, TC Hudson of the Nautical Almanac Office approached the Burroughs Adding Machine Company with a request to design a difference machine that worked in the sixties number system. The tables to be calculated by Hudson listed angles in degrees, minutes and seconds. Even the first Babbage machine was able to deal with different number systems of different bases by simply swapping the number rollers, as the English currency at that time was not based on pence , shilling and pound : 12 pence = 1 shilling, 20 shilling = 1 pound. Hudson got a keyboard-operated accounting machine for his work, which could calculate in both the tens and sixty systems and could also subtract directly, i.e. it did not need to form a complement before adding. However, the machine could only perform an addition or subtraction, which is why the expression of the first difference was run through the machine one more time in order to calculate the actual function value. By cleverly clamping the printout, the result of the first calculation was overprinted by the input of the first difference in the second calculation step, which made incorrect entries obvious. The machine was exhibited at the Napier Tercentenary Exhibition in Edinburgh in 1914 . For daily use, two of the machines were coupled one after the other via their printout, so that the first machine added the second differences to the first and the second machine added the first difference to the result. Burroughs later added an additional register for the first difference in the machines so that one accounting machine could be used as a difference machine with two differences.

Leslie John Comrie

Leslie John Comrie carried on T. C. Hudson's idea of ​​converting standard office machines to differential machines with minimal changes: he examined and described every new machine in detail. Comrie mainly used the difference machines to control tables.

Brunsviga Dupla

In 1928 Comrie presented the Brunsviga- Dupla, a calculating machine with an intermediate register, as a difference machine. He practically wrote an instruction manual for the calculating machine as a difference machine, explaining how one should use the machine's own intermediate register. The machine is very rare, nowadays the calculation methods can be most easily understood through simulations.

Hollerith tab machine

It was Comrie who, during the disassembly of a new booking machine, the Hollerith Tabulating Machine , discovered the mechanical registers hidden in the machine for the subtotals of the booking accounts and described the use of the machine as a difference machine. The tabulation machines could be used as a printing difference machine without any modifications.

Triumphator type C1.
Four of these machines formed the basis for the difference machine built by Thompson.

Alexander John Thompson

For the calculation of his 20-digit logarithm tables published between 1924 and 1952, Alexander John Thompson looked in vain for a machine that could calculate with four or five differences. He mechanically coupled four Triumphator Type C computing machines ( Sprossenradmaschinen , also called Odhner machines after Willgodt Theophil Odhner ) one behind the other so that the contents of the result register of the rear machine could be transferred to the Sprossenräder of the front machine. The settings of the sprocket wheels of the front machine could also be transferred to the results register of the rear machine. In addition, the number of sprocket wheels on each machine was expanded from 9 to 13 digits, and the result register from 13 to 18 digits.

functionality

Difference machines are used to calculate sequences of numbers . The difference is the numerical distance between two adjacent elements of the sequence. These differences in turn form a consequence. If the difference is calculated again from these first differences, one speaks of the second difference.

In general, in a sequence based on an nth order polynomial, the nth difference is constant. This property can be used to the effect that for the construction of the sequence, first the nth, then the n-1th etc. differences are formed by addition until the actual sequence has been calculated. Of course you have to know the starting values ​​for the individual differences for this procedure. The one-time determination of the starting values ​​is usually much easier than calculating all elements of the sequence.

The most common use of the difference machines was the interpolation of values ​​between known support points: For a function, the function values ​​were calculated with the required accuracy at a greater distance than for the desired table. The intermediate values ​​were obtained by calculating an nth order polynomial using the calculated support points. I.e. the intermediate values ​​were only approximated by the polynomial. With an appropriate choice of the distances between the support position and sufficient accuracy of the calculation steps, the calculated and interpolated table values ​​cannot be distinguished. Interpolation was also used before difference machines became available, only that the calculation then had to be performed by humans. Here, too, the computational effort for the interpolation was significantly lower than for the calculation of the support points.

It was shown early on that interesting application possibilities arise for the difference machine if the nth difference is not constant. Charles Babbage provided that the operator could adjust the highest difference before each calculation step. This is particularly easy to do in the Thompson machine. Since the difference machine can actually only add, Charles Babbage saw in his difference machine No. 1 provides a feedback of the sequence to the highest difference. It should be noted that he could vary the value of the digits by powers of 10 during the feedback. With this trick the difference machine could multiply and z. B. Calculate the sine function directly.

Legend has it that Charles Babbage dedicated himself to printing difference machines because there were so many errors in the available log tables. In the meantime, the number of errors in the log tables has been objectified. Although it was possible to track the errors, it was possible to prove which board maker wrote off whom, but overall the number of errors, especially when taking the correction notes into account, was so low that this can be seen as an argument for embarrassment for the construction of such a complex machine . For the Scheutz No. 2 machine, there is extensive evidence of the functional quality and susceptibility to repair: The machine was neither faultless, nor was it cheaper to maintain than a human calculator.

Appreciation

At the time of the invention of the difference machine by Charles Babbage, the machine reproduced a calculation process that was carried out manually up to that point in time. Charles Babbage succeeded in 1822 in building and demonstrating a functioning difference machine (No. 0) for second order differences; unfortunately the machine went down. It is very likely that individual parts were used in the demonstrator for Difference Machine No. 1, but certainly in the fragments of the difference machine no. 1, installed. The machines Babbage designed were manufacturable in his day and would have worked. Through the numerous publications about his difference machines and advertising for difference machines that were designed and built using his articles, he inspired generations of mathematicians, engineers and hobbyists to try their hand at calculating automation. The big breakthrough of the difference machines came with the use of Christel Hamann's machine for the tables of Bauschinger and Peters. However, the use of the difference machine reached its climax and conclusion in the four-difference machine by Alexander John Thompson and his calculation of the 20-digit log tables between 1924 and 1952.

List of difference machines

Difference machines
inventor builder Surname machine start of building completion Differences Make arithmetic unit Make printer commitment Today's location Functionality source
Johann Helfrich Müller description --- --- --- --- ---
Charles Babbage Charles Babbage No. 0 functioning 1820 1822 2 6th --- Charles Babbage --- ---
Charles Babbage Joseph Clement No. 1 Demonstrator 1832 2 --- --- Science Museum, London Yes
Charles Babbage Roberto Guatelli No. 1 Demonstrator, replica circa 1981 2 --- --- Canada Science and Technology Museum, Ottawa, Canada Yes
Charles Babbage Doron Swade No. 2 functioning 1991/2000 7th 31 Science Museum, London Yes
Charles Babbage Doron Swade No. 2 functioning 2008 7th 31 Computer History Museum, ca. Yes
George Scheutz Edvard Scheutz Demonstrator 1837 1843 3 5 --- --- Tekniska museet, Stockholm No
George Scheutz Edvard Scheutz, Johan Wilhelm Bergström Scheutz No. 1 functioning 1851 1853 4th 15th 8th Dudley Observatory, Schenectady, NY National Museum of American History, Smithonian Institution, Washington DC Yes
George Scheutz Edvard Scheutz, Bryan Donkin Scheutz No. 2 functioning 1857 1859 4th 15th 8th General Register Office, Sommerset House, London Science Museum, London Yes
Martin Wiberg Martin Wiberg functioning 1860 4th 15th Tekniska museet, Stockholm Yes
Alfred Deacon Alfred Deacon functioning 1862 3 20th
George Barnard Grant George Barnard Grant Demonstrator 1870 1871 --- --- ---
George Barnard Grant George Barnard Grant Grant's Difference Engine conditionally functional 1874 1876 --- --- ---
George Barnard Grant George Barnard Grant (?) Grant's Difference Engine functioning 1876 ​​or later Provident Mutual Life Insurance Company unknown Yes
Christel Hamann Christel Hamann functioning 1909 2 16 Bauschinger, Peters --- ---
TC Hudson Burroughs Adding Machine Company functioning 1912 2
Brunsviga Brunsviga Dupla functioning 1928 2 16 --- Deutsches Museum , Munich; Braunschweigisches Landesmuseum (currently not on display)
Hollerith Tabulating Machine functioning
Alexander John Thompson Alexander John Thompson, 4 times Triumphator Type C functioning 1924-1950 4th 13 --- Alexander John Thompson Dept. of Statistical Science, University College London Yes

literature

• Michael Lindgren: Glory and failure . The difference engines of Johann Müller, Charles Babbage and Georg and Edvard Scheutz (=  Stockholm papers in history and philosophy of technology . Volume 2017 ). 2nd Edition. MIT Press, 1990, ISBN 0-262-12146-8 (also Linköping University dissertation (Linköping studies in arts and science, Volume 9, 1987)).
• Uta C. Merzbach : Georg Scheutz and the first printing calculator (=  Smithsonian Studies in History and Technology . Volume 36 ). Smithsonian Institution Press, Washington, DC 1977 ( online [PDF; 30.9 MB ; accessed on May 5, 2012]).
• Bernhard Dotzler (Hrsg.): Babbages Rechen-Automate. (=  Computer culture . Band IV ). Springer, Vienna / New York 1996, ISBN 3-211-82640-8 .
• Doron Swade: The Cogwheel Brain . Charles Babbage and the Quest to Build the First Computer. 1st edition. Little, Brown & Co., 2000 (reprint under The Difference Engine at Pinguin).

Commons : Difference engines  - collection of images, videos, and audio files

Individual evidence

1. ^ Martin Campbell-Kelly, William Aspray: Computer . A History of the Information Machine (=  The Sloan Technology Series ). 1st edition. Basic Books, New York 1996, ISBN 0-465-02990-6 .
2. ^ Calculation of differences . BG Teubner, Leipzig 1896, chap. 4 ( online [accessed September 8, 2013]).
3. a b c d Dionysius Lardner: Babbage's calculating engine . In: The Edinburgh Review . tape 59 , July 1834 (Also available in Campbell-Kelly (1989). Pp. 118-186 and as a German translation in Dotzler (1996) , section 12.).
4. a b c d George Scheutz, Edward Scheutz: Specimen Tables . Calculated, Stereomoulded, and Printed by the Swedish Calculating Machine. Longman, Brown, Green, Longmans, and Roberts, London 2010, ISBN 978-1-167-03892-1 ( in Google Books or as a PDF file (26 MB) [accessed September 20, 2012] First edition: 1857, reprint ).
5. a b c d Julius Bauschinger, Jean Peters: Logarithmic trigonometric tables with eight decimal places . The logarithms of all numbers from 1 to 200,000 and the logarithms of the trigonometric functions for each sexagesimal second of the quadrant. 1st edition. tape 1/2 . Verlag von Wilhelm Engelmann, Leipzig 1910 ( online [accessed on September 17, 2012] With the support of the Royal Prussian Academy of Science in Berlin and the Imperial Academy of Science in Vienna (Treitl Foundation)).
6. a b Julius Bauschinger, Jean Peters: Logarithmic trigonometric tables with eight decimal places . The logarithms of all numbers from 1 to 200,000 and the logarithms of the trigonometric functions for each sexagesimal second of the quadrant. 1st edition. tape 2/2 . Verlag von Wilhelm Engelmann, Leipzig 1910 ( online [accessed September 17, 2012]).
7. Leslie John Comrie: On the Application of the Brunsviga-Dupla Calculating Machine to Double Summation with Finite Differences . In: Monthly Notices of the Royal Astronomical Society . tape 88 , March 1928, p. 447–459, plates 4 & 5 , bibcode : 1928MNRAS..88..447C .
8. ^ Leslie John Comrie: Modern Babbage Machines . A typescript of Comrie's Modern Babbage Machines. In: Bulletin of the Office Machinery Users' Association Limited . 1933.
9. Martin Campbell-Kelly, Mary Croarken, R. Flood, Eleanor Robson (Eds.): The History of Mathematical Tables . From Sumer to Spreadsheets. Oxford University Press, Oxford 2003, ISBN 0-19-850841-7 .
10. a b c d Alexander John Thompson: Logarithmetica britannica being a standard table of logarithms to twenty decimal places . Numbers 10,000 to 50,000 together with general introduction. tape 1/2 . Cambridge 1952.
11. a b Alexander John Thompson: Logarithmetica britannica being a standard table of logarithms to twenty decimal places . Numbers 50,000 to 100,000. tape 2/2 . Cambridge 1952.
12. Otto Weber: A computer of the 18th century . The calculating machine of the master builder Müller. In: Photorin . Announcements from the Lichtenberg Society. No. 3 , December 1980, p. 13-23 .
13. ^ A b c Johann Helfrich von Müller: Description of his newly invented calculating machine . According to their shape, their use and benefits. Ed .: Philipp Engel Klebstein. Varrentrapp Sohn and Wenner, Frankfurt 1786, OCLC 633555891 (Augsburg State and City Library, call number : H 1594.).
14. a b c Ralph Bülow: A draft for a difference machine from 1784 . In: Sudhoff's archive . tape 73 , no. 2 , 1989, pp. 219-222 .
15. ^ Charles Babbage: A Note Respecting the Application of Machinery to the Calculation of Astronomical Tables . In: Memoirs of the Astronomical Society . tape 1 , 1822, p. 309 (Also available in Campbell-Kelly (1989), and as a German translation in Bernhard Dotzler (1996)).
16. ^ A b Charles Babbage: On the Applications of Machinery to the Purpose of Calculating and Printing Mathematical Tables . In: Parliamentary Papers . tape 15 , 1823, pp. 9-14 (Originally letter to Sir Humphry Davy, 1822. Also available in Campbell-Kelly (1989), and as a German translation in Bernhard Dotzler (1996)).
17. ^ A b c d Nicholas K. Taylor: Charles Babbage's Mini-Computer . Difference Engine No. 0. In: Bulletin . The Institute of Mathematics and its Applications. tape 28 , 6/7/8 (June / July / August). Southend-on-Sea, Essex 1992, pp. 112-114 .
18. ^ A b Garry John Tee: More about Charles Babbage's Difference Engine No. 0 . In: Bulletin . The Institute of Mathematics and its Applications. tape 30 , 9/10 (September / October). Southend-on-Sea, Essex 1994, p. 134-137 .
19. ^ Charles Babbage: The science of number reduced to mechanism . 1821 (Unpublished during Babbage's lifetime. Reprinted in Campbell-Kelly (1989)).
20. a b c Denis Roegel: Prototype Fragments from Babbage's First Difference Engine . In: IEEE Annals of the History of Computing . tape 31 , 2 (April – June), 2009, ISSN  1058-6180 , pp. 70–75 , doi : 10.1109 / MAHC.2009.31 ( online [accessed August 12, 2012]).
21. a b Ole Immanuel Franksen: Mr. Babbage, the difference engine, and the problem of notation . An account of the origin of recursiveness and conditionals in computer programming. In: International Journal of Engineering Science . tape 19 , no. 12 , 1981, p. 1657-1694 , doi : 10.1016 / 0020-7225 (81) 90158-0 .
22. ^ Allan G. Bromley: Review of OI Franksen's 1981 article on Charles Babbage's Difference Engine . In: IEEE Annals of the History of Computing . tape 5 , no. 4 , 1987, pp. 411-415 , doi : 10.1109 / MAHC.1983.10092 .
23. ^ A b C. JD Roberts: Babbage's Difference Engine No. 1 and the Production of Sine Tables . In: Werner Buchholz (Ed.): IEEE Annals of the History of Computing . tape 9 , no. 2 , 1987, pp. 210-212 , doi : 10.1109 / MAHC.1987.10020 .
24. a b c Garry John Tee: The Heritage of Charles Babbage in Australasia . In: Annals of the History of Computing . tape 5 , no. 1 , 1983, p. 45-60 , doi : 10.1109 / MAHC.1983.10006 .
25. a b c d Charle Babbage: Passages from the Life of a Philosopher . Longman, Green, Longman, Roberts & Green, London 1864 ( online [accessed August 15, 2012]).
26. a b c d I. Bernard Cohen: Babbage and Aiken . with Notes on Henry Babbage's Gift to Harvard, and to other Institutions, of a Portion of His Father's Difference Engine. In: IEEE Annals of the History of Computing . tape 10 , 3 (July – September), 1988, pp. 171-193 , doi : 10.1109 / MAHC.1988.10029 .
27. ^ Allan G. Bromley: The Evolution of Babbage's Calculating Engines . In: IEEE Annals of the History of Computing . tape 9 , no. 2 , 1987, pp. 113-136 , doi : 10.1109 / MAHC.1987.10013 .
28. ^ NS Dodge: Charles Babbage eulogy . In: IEEE Annals of the History of Computing . tape 22 , no. 4 , 2000, pp. 22-43 , doi : 10.1109 / MAHC.2000.887987 (annotated reprint from the Smithonian Report of 1873).
29. ^ Gerard O'Regan: A Brief History of Computing. Springer Science & Business Media, ISBN 978-1-4471-2359-0 .
30. Doron D. Swade: The mechanical computer of Charles Babbage. In: Spectrum of Science. 4, 1993, pp. 78-84.
31. Completion of the printer: 2000
32. Website of the museum : "An identical Engine completed in March 2008 is on display at the Computer History Museum"
33. Website of the museum : "The Babbage Difference Engine No. 2 was on Display from May 2008 till January 2016"
34. Second Original
35. Michael Lindgren: Glory and failure . The difference engines of Johann Müller, Charles Babbage and Georg and Edvard Scheutz (=  Stockholm papers in history and philosophy of technology . Volume 2017 ). 2nd Edition. MIT Press, 1990, ISBN 0-262-12146-8 (also Linköping University dissertation (Linköping studies in arts and science, Volume 9, 1987)).
36. ^ Dionysius Lardner: Babbage's calculating engine . In: Martin Campbell-Kelly (Ed.): The Works of Charles Babbage . The Difference Engine and Table Making. tape 2 . William Pickering, London 1989, ISBN 1-85196-005-8 , pp. 118-186 (reprinted from The Edinburgh Review of 1834).
37. Exhibit visible here
38. Uta C. Merzbach: Georg Scheutz and the first printing calculator . In: Smithsonian Studies in History and Technology . tape 36 . Smithsonian Institution Press, City of Washington 1977, LCCN  76-015379 ( online [PDF; 30.9 MB ; accessed on September 10, 2012]).
39. George G. Stokes, WH Miller, Charles Wheatstone, R. Willis: Report of a Committee appointed by the Council to examine the Calculating Machine of M [essrs] Scheutz . 1855, p. 499-509 .
40. ^ Edwards Park: What a difference the Difference Engine made . From Charles Babbage's calculator emerged today's computer. In: Smithsonian magazine . February 1996 ( online [accessed August 5, 2012]). online ( Memento of the original from November 21, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
41. ^ William Gravatt, Companion to the Barometer, Mountain Barometer Tables; Calculated and Stereotyped by Messrs. Scheutz's Calculating Machine No. 2 and Printed by Machinery. London 1859.
42. Mathieum Chasles, Delaunay: Rapport sur la machine a calculer presentee par M. Wiberg . In: Competes Rendus . Hebdomadaires des seances de L'academie des Sciences. tape 56 , no. 1 . Mallet-Bachelier, Paris 1863, p. 330–339 ( online [accessed September 16, 2012]).
43. ^ Martin Wiberg: Tables de logarithmes, calculées et imprimées au moyen de la machine à calculer . Compagnie anonyme de Forsete, Stockholm 1876.
44. ^ A b Raymond Clare Archibald : Martin Wiberg, his Table and Difference Engine . In: Mathematical Tables and Other Aids to Computation . tape 2 , no. October 20 , 1947, p. 371–374 ( online [PDF; 561 kB ] review).
45. ^ A b c George Bernard Grant: On a New Difference Engine . In: American Journal of Science . Third Series. August 1871, p. 113–117 ( online [PDF; 90.8 MB ]).
46. ^ A b Michael R. Williams: A History of Computing Technology . 2nd Edition. IEEE Computer Science Press, Los Alamitos, ca. 1995, ISBN 0-8186-7739-2 .
47. ^ A b Leslie John Comrie: The Nautical Almanac Office Burroughs Machine . In: Monthly Notices of the Royal Astronomical Society . tape 92 , April 1932, p. 523-541, plates 5 & 6 , bibcode : 1932MNRAS..92..523C .
48. ^ TC Hudson: HM Nautical Almanac Office Anti-Difference Machine . In: Ellice Martin Horsburg (Ed.): The Tercentenary Exhibition of Napier Relics . 1914: Edinburgh, Scotland. G. Bell and Sons, Ltd., Royal Society of Edinburgh, London 1914, p. 127-131 ( online [accessed September 30, 2012]).
49. ^ Leslie John Comrie: Modern Babbage Machines . A typescript of Comrie's Modern Babbage Machines. In: Bulletin of the Office Machinery Users' Association Limited . 1933, p. 29 .
50. Flash simulation of the Brunsviga-Dupla
51. ^ A b Leslie John Comrie: The application of the Hollerith tabulating machine to Brown's tables of the moon . In: Monthly Notices of the Royal Astronomical Society . tape 92 , April 1932, p. 694-707, plates 11-13 , bibcode : 1932MNRAS..92..694C .
52. ^ Stephan Weiss: Difference Engines in the 20th Century . In: Proceedings 16th International Meeting of Collectors of Historical Calculating Instruments . Leiden September 2010 ( online [PDF; 859 kB ; accessed on September 8, 2013]).