Difference engine: Difference between revisions

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{{Short description|Automatic mechanical calculator}}

{{For|the novel by William Gibson and Bruce Sterling|The Difference Engine}}

[[File:Babbage Difference Engine.jpg|thumb|The London [[Science Museum (London)|Science Museum]]'s difference engine, the first one actually built from Babbage's design. The designIt has the same precision on all columns, butexcept in calculating polynomials, the precision on the higher-order columns could be lower.]]

A '''difference engine''' is an automatic [[mechanical calculator]] designed to tabulate [[polynomial]] functions. It was designed in the 1820s, and was first created by [[Charles Babbage]]. The name ''difference engine'' is derived from the method of [[dividedfinite differences]], a way to interpolate or tabulate functions by using a small set of polynomial co-efficients. Some of the most common [[mathematical function]]s used in engineering, science and navigation are built from [[logarithm]]ic and [[trigonometric functions]], which can be [[Taylor series|approximated]] by polynomials, so a difference engine can compute many useful [[Mathematical table|tables]].

== History ==

{{Wikisource|Astronomische Nachrichten/Volume 46/On Mr. Babbage's new machine for calculating and printing mathematical and astronomical tables}}

[[File:LondonScienceMuseumsReplicaDifferenceEngine.jpg|thumb|Close-up of the londonLondon Science Museum's difference engine showing some of the number wheels and the sector gears between columns. The sector gears on the left show the double-high teeth very clearly. The sector gears on the middle-right are facing the back side of the engine, but the single-high teeth are clearly visible. Notice how the wheels are mirrored, with counting up from left-to-right, or counting down from left-to-right. Also notice the metal tab between "6" and "7". That tab trips the carry lever in the back when "9" passes to "0" in the front during the add steps (Step 1 and Step 3).]]

The notion of a [[mechanical calculator]] for mathematical functions can be traced back to the [[Antikythera mechanism]] of the 2nd century BC, while early modern examples are attributed to [[Blaise Pascal|Pascal]] and [[Gottfried Leibniz|Leibniz]] in the 17th century.

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Charles Babbage began to construct a small difference engine in {{circa|1819}}<ref>{{Cite book |url=https://books.google.com/books?id=tXBVAgAAQBAJ&pg=PT22 |title=It Began with Babbage: The Genesis of Computer Science |last=Dasgupta |first=Subrata |date=2014 |publisher=Oxford University Press |isbn=978-0-19-930943-6|page=22}}</ref> and had completed it by 1822 (Difference Engine 0).<ref name=":0">{{Cite book |title=The Turing Guide |author-link1=Jack Copeland |last1=Copeland |first1=B. Jack |author-link2=Jonathan Bowen |last2=Bowen |first2=Jonathan P. |author-link3=Robin Wilson (mathematician) |last3=Wilson |first3=Robin |last4=Sprevak |first4=Mark |date=2017 |publisher=[[Oxford University Press]] |isbn=9780191065002 |page=251 |title-link=The Turing Guide }}</ref> He announced his invention on 14 June 1822, in a paper to the [[Royal Astronomical Society]], entitled "Note on the application of machinery to the computation of astronomical and mathematical tables".<ref>{{cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |author-link2=Edmund F. Robertson |date=1998 |url=http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.html |title=Charles Babbage |work=MacTutor History of Mathematics archive |publisher=School of Mathematics and Statistics, University of St Andrews, Scotland |access-date=2006-06-14 |archive-url=https://web.archive.org/web/20060616002258/http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.html |archive-date=2006-06-16 }}</ref> This machine used the decimal number system and was powered by cranking a handle. The [[British government]] was interested, since producing tables was time-consuming and expensive and they hoped the difference engine would make the task more economical.<ref name="Campbell-Kelly 2004">{{cite book |title=Computer: A History of the Information Machine 2nd ed. |last=Campbell-Kelly |first=Martin |publisher=Westview Press |date=2004 |isbn=978-0-8133-4264-1 |___location=Boulder, Colorado |author-link=Martin Campbell-Kelly |url-access=registration |url=https://archive.org/details/computerhistoryo02edcamp }}</ref>

In 1823, the British government gave Babbage £1700 to start work on the project. Although Babbage's design was feasible, the metalworking techniques of the era could not economically make parts in the precision and quantity required. Thus the implementation proved to be much more expensive and doubtful of success than the government's initial estimate. According to the 1830 design for Difference Engine No. 1, it would have about 25,000 parts, weigh 4 [[ton]]s,<ref>{{Cite web |title=The Engines {{!}} Babbage Engine |url=https://www.computerhistory.org/babbage/engines/ |publisher=Computer History Museum |access-date=2022-07-10 }}</ref> and operate on 20-digit numbers by sixth-order differences. In 1832, Babbage and [[Joseph Clement]] produced a small working model (one-seventh of the plan),<ref name=":0" /> which operated on 6-digit numbers by second-order differences.<ref name=":1">{{Cite book |url=https://books.google.com/books?id=QqrItgm351EC&pg=PA204 |title=A Brief History of Computing |last=O'Regan |first=Gerard |date=2012 |publisher=Springer Science & Business Media |isbn=978-1-4471-2359-0|page=204}}</ref><ref name=":2">{{Cite book |url=https://books.google.com/books?id=JckCvpOQDOoC&pg=PP1 |title=The Philosophical Breakfast Club: Four Remarkable Friends Who Transformed Science and Changed the World |last=Snyder |first=Laura J. |date=2011 |publisher=Crown/Archetype|isbn=978-0-307-71617-0 |pages=192, 210, 217 }}</ref> [[Lady Byron]] described seeing the working prototype in 1833: "We both went to see the thinking machine (or so it seems) last Monday. It raised several Nos. to the 2nd and 3rd powers, and extracted the root of a Quadratic equation."<ref>{{Cite book |title=Ada, the Enchantress of Numbers |last1=Toole |first1=Betty Alexandra |last2=Lovelace |first2=Ada |date=1998 |publisher=Strawberry Press |isbn=978-0912647180 |___location=Mill Valley, California |oclc=40943907 |page=[https://archive.org/details/adaenchantressof00tool/page/38 38] |url=https://archive.org/details/adaenchantressof00tool/page/38 }}</ref> Lady Byron's daughter [[Ada Lovelace]] would later become fascinated with and work on creating the first computer program intended to solve Bernoulli's equation utilizing the difference engine. Work on the larger engine was suspended in 1833.

By the time the government abandoned the project in 1842,<ref name=":2" /><ref>{{Cite book |url=https://books.google.com/books?id=UmNJAAAAYAAJ&pg=PA387 |title=A History of the Royal Society: With Memoirs of the Presidents |last=Weld |first=Charles Richard |date=1848 |publisher=J. W. Parker |pages=387–390 }}</ref> Babbage had received and spent over £17,000 on development, which still fell short of achieving a working engine. The government valued only the machine's output (economically produced tables), not the development (at unpredictable cost) of the machine itself. Babbage refused to recognize that predicament.<ref name="Campbell-Kelly 2004" /> Meanwhile, Babbage's attention had moved on to developing an [[analytical engine]], further undermining the government's confidence in the eventual success of the difference engine. By improving the concept as an analytical engine, Babbage had made the difference engine concept obsolete, and the project to implement it an utter failure in the view of the government.<ref name="Campbell-Kelly 2004" />

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The incomplete Difference Engine No. 1 was put on display to the public at the [[1862 International Exhibition]] in [[South Kensington]], London.<ref>{{Cite book |url=https://books.google.com/books?id=aDJRAAAAYAAJ |title=Cyclopaedia of useful arts, mechanical and chemical, manufactures, mining and engineering: in three volumes, illustrated by 63 steel engravings and 3063 wood engravings |last=Tomlinson |first=Charles |date=1868 |publisher=Virtue & Co. |page=136 }}</ref><ref>{{Cite book |url=https://archive.org/details/bub_gb_z7MpAAAAYAAJ|title=Official catalogue of the industrial department |date=1862 |page=[https://archive.org/details/bub_gb_z7MpAAAAYAAJ/page/n70 49] }}</ref>

Babbage went on to design his much more general analytical engine, but later produceddesigned an improved "Difference Engine No. 2" design (31-digit numbers and seventh-order differences),<ref name=":1" /> between 1846 and 1849. Babbage was able to take advantage of ideas developed for the analytical engine to make the new difference engine calculate more quickly while using fewer parts.<ref>{{cite book |title=The Philosophical Breakfast Club |last=Snyder |first=Laura J. |publisher=Broadway Brooks |date=2011 |isbn=978-0-7679-3048-2 |___location=New York |url-access=registration |url=https://archive.org/details/philosophicalbre0000snyd}}</ref><ref>{{Cite book |title=The Dawn of Innovation: The First American Industrial Revolution |last=Morris |first=Charles R. |url=https://books.google.com/books?id=hRQBAwAAQBAJ&pg=PA63 |date=October 23, 2012 |publisher=PublicAffairs |isbn=9781610393577 |page=63 }}</ref>

=== Scheutzian calculation engine ===

[[File:Difference engine Scheutz.jpg|thumb|upright=1.8|Per Georg Scheutz's third difference engine, in the [[Science Museum, London|Science Museum]], London]]

Inspired by Babbage's difference engine in 1834, the Swedish inventor [[Per Georg Scheutz]] built several experimental models. In 1837 his son Edward proposed to construct a working model in metal, and in 1840 finished the calculating part, capable of calculating series with 5-digit numbers and first-order differences, which was later extended to third-order (1842). In 1843, after adding the printing part, the model was completed.

In 1851, funded by the government, construction of the larger and improved (15-digit numbers and fourth-order differences) machine began, and finished in 1853. The machine was demonstrated at the [[Exposition Universelle (1855)|World's Fair in Paris, 1855]] and then sold in 1856 to the [[Dudley Observatory]] in [[Albany, New York]]. Delivered in 1857, it was the first printing calculator sold.<ref name=":3">{{Cite book|url=https://archive.org/details/bub_gb_Ut1wgt6kSBEC|title=Specimens of Tables, Calculated, Stereomoulded, and Printed by Machinery|last1=Scheutz|first1=George|last2=Scheutz|first2=Edward|date=1857 |publisher=Whitnig |pages=VIII–XII, XIV–XV, 3}}</ref><ref name="SI">{{cite web |title=Scheutz Difference Engine |url=https://americanhistory.si.edu/collections/search/object/nmah_997042 |website=Smithsonian National Museum of American History |access-date=June 14, 2019}}</ref><ref name=":6" /> In 1857 the British government ordered the next [[Per Georg Scheutz|Scheutz's]] difference machine, which was built in 1859.<ref>{{Cite book |url=https://archive.org/details/differenceengine00doro|url-access=registration|title=The Difference Engine: Charles Babbage and the Quest to Build the First Computer|last=Swade|first=Doron|date=2002-10-29 |publisher=Penguin Books |isbn=9780142001448 |pages=[https://archive.org/details/differenceengine00doro/page/4 4], 207 }}</ref><ref>{{Cite book|url=https://books.google.com/books?id=jlmVKZ1psCkC&pg=PA37|title=The Universal Machine: From the Dawn of Computing to Digital Consciousness|last=Watson|first=Ian|date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-28102-0|pages=37–38}}</ref> It had the same basic construction as the previous one, weighing about {{Convert|10|-Lcwt|lb kg|lk=on|abbr=on}}.<ref name=":6">{{Cite book |title=First Printing Calculator |last1=Merzbach |first1=Uta C. |author-link=Uta Merzbach |last2=Ripley |first2=S. Dillon |last3=Merzbach |first3=Uta C. |pages=8–9, 13, 25–26, 29–30 |citeseerx=10.1.1.639.3286 }}</ref>

=== Others ===

[[Martin Wiberg]] improved Scheutz's construction ({{circa|1859}}, his machine has the same capacity as Scheutz's: 1530-digit and fourthsixth-order) but used his device only for producing and publishing printed tables (interest tables in 1860, and [[logarithm]]ic tables in 1875).<ref>{{cite journal |first=Raymond Clare |last=Archibald |title=Martin Wiberg, His Table and Difference Engine |journal=Mathematical Tables and Other Aids to Computation |date=1947 |volume=2 |issue=20 |pages=371–374 |url=https://www.ams.org/journals/mcom/1947-02-020/S0025-5718-47-99566-5/S0025-5718-47-99566-5.pdf }}</ref>

Alfred Deacon of London in {{circa|1862}} produced a small difference engine (20-digit numbers and third-order differences).<ref name=":3" /><ref name=":4">{{Cite book|at=[https://books.google.com/books?id=O170gWPZ7M8C&pg=PA136 pp. 132–136]|title=The History of Mathematical Tables: From Sumer to Spreadsheets|last=Campbell-Kelly|first=Martin|date=2003|publisher=OUP Oxford|isbn=978-0-19-850841-0|title-link= The History of Mathematical Tables}}</ref>

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* [[Allan G. Bromley]]

* [[Johann Helfrich von Müller]]

* [[Logical machine]]

* [[Martin Wiberg]]

* [[Pinwheel calculator]]

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== Further reading ==

{{Refbegin|}}

* {{cite book |title=The Philosophical Breakfast Club: Four Remarkable Friends Who Transformed Science and Changed the World |last=Snyder |first=Laura J. |publisher=Broadway |date=2011 |isbn=978-0-7679-3048-2 |url-access=registration |url=https://archive.org/details/philosophicalbre0000snyd }}

* {{cite book |url=http://ed-thelen.org/bab/bab_tech.html |title=Charles Babbage's Difference Engine No. 2 – Technical Description |last=Swade |first=Doron |date=September 1996 |publisher=[[National Museum of Science and Industry]] |series=Science Museum Papers in the History of Technology No 5 |___location=London |access-date=2009-01-11}}

* {{cite book |title = The Difference Engine: Charles Babbage and the Quest to Build the First Computer |last = Swade |first =Doron Doron|publisher = Penguin (reprint) |date=2002 2002|isbn = 978-0-14-200144-8 |author-link = Doron Swade |url = https://archive.org/details/differenceengine00doro}}

* {{cite book |title=The Cogwheel Brain |last=Swade |first=Doron |publisher=Abacus |date=2001 |isbn=978-0-349-11239-8 }}

* {{cite video |people=Doron Swade, [[Nathan Myhrvold]] |title=Myhrvold & Swade Discuss Babbage's Difference Engine |url=https://www.youtube.com/watch?v=p1sEowi1Txc |archive-url=https://ghostarchive.org/varchive/youtube/20211211/p1sEowi1Txc |archive-date=2021-12-11 |url-status=live |medium=lecture: [[Len Shustek]], intro; Doron Swade @7:35, Nathan Myhrvold @36:25; discussion @46:45 |publisher=Computer History Museum |date=June 10, 2008 |access-date=2009-11-06 }}{{cbignore}}

* {{cite book |chapter-url=https://books.google.com/books?id=O170gWPZ7M8C |title=The History of Mathematical Tables: From Sumer to Spreadsheets |last1=Campbell-Kelly |first1=Martin |date=2003 |publisher=OUP Oxford |others=Michael R. Williams |isbn=9780198508410 |chapter=Difference engines: from Müller to Comrie }}

*Hollings, C., Martin, U., & Rice, A. C. (2018). Ada lovelace: The making of a computer scientist. Bodleian Library.{{Refend}}

{{Refend}}

== External links ==

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* Links to videos about Babbage DE 2 and its construction: {{cite web |title=Computer Histories: To Learn More |url=http://www.computerhistories.org/ToLearnMore.html |website=www.computerhistories.org|at=Topic 5 - Computers in the Steam Era (Not Hackers But Clackers)}}

{{Calculator navbox}}

{{Authority control}}

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[[Category:Articles containing video clips]]

[[Category:Charles Babbage]]

[[Category:CollectionsCollection of the Science Museum, London]]

[[Category:Computer-related introductions in the 19th century]]

[[Category:English inventions]]