Babbage Difference Analytical Engine

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Chapt er 2Di f f erence andAnal yt i cal Engi nes

ntroducti on

Thedevel opment of asuccessful comput i ngmachi ne requi res theprovi si on of mechani sm or a t l ea s t two basi c f unct i ons- t hestorage and ari thmet i c mani pul at i on of number s , and some controlmechani smwhereby a s er i es of ari thmet i c operat i ons may becombi ned t o produce the r e s u l t of a desi red c a l c u l a t i on . Theprevi ous chapter descri bed the devel opment of such mechani sms -the commerci al l y s u c c e s s f u l machi nes of Thomas de Col mar i n the1820s bei ng the f i r s t to combi ne a pr a c t i c al desi gnw th an e f f e c t i v emethod of manuf acture . I n these machi es the control f unct i on wasprovi ded by the human operator .Devel opment s were sl ow however , and i t was not unt i l the 1880sand 1890s t h a t " s c i e n t i f i c " machi nes ( capabl e of mul t i pl i cat i on anddi vi si on) f ol l ow ng the desi gns of de Colmar or Odhner and"commerci al " machi nes ( capabl e of addi t i on, or addi t i on andsubt racti on onl y) by Fel t andTarrant, Burroughs, and others begant o appear i n quant i ty Theneed f o r thesewas spurred bythedemandsof the l a r ge r busi nesses bred by t h e I ndus t r i a l Revol ut i on I n turn theI ndust ri al Revol ut i on made possi bl e the economc mass product i onof t he cal cul ators t hemel ves .Devi ces f or t he automat i c control of mechani sms have amuchl onger hi s t o r y , l eadi ng back t o t he anci ent Greek c i v i l i z a t i o n s .


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Automati c mechani sms were extensi vel y devel oped i n the greatastronomcal cl ocks of the Renaissance and the automata of t heei ghteenthcenturyTwodevices of par t i cul ar importance t oour s t o r y ,thepi nbarrel musi cboxandthepunched cardJ acquard l oom werewel l est abl i shed by the ear l y ni neteenth century O consi der abl eimportance, but i n a more abst r act i nt el l ect ual sense, was t hedevel opment byStephensonof t he bal l governor f or thesteamengi neandthe i deaof feedback that i t embodiesBythe ear l y ni neteenth cent ur y, t her ef or e, the basi c mechani sm

andi deas exi stedfromwhi chanautomati c cal cul at i ngmachinecoul dbedevel oped Thi swas doneby t he Engl i shmathemati ci anChar l esBabbage, whodeveloped, si ngl e- handedl y, most of the basi c i deasi nherent i n the l ogi cal desi gn of modern di gi t al computers-ani nt el l ect ual tour de fo r ce sel domequal l ed i n the hi st or y of sci enceandtechnol ogy Babbage' s i deas wereembodi edi n thedesi gnof twocal cul at i ng machi nes, t he D fference Engi ne and t he Anal yt i calEngi ne, whi chformthemai nt opi cs of t h i s chapter .

harl esBabbageCar l es Babbagewas born i n south Londonon December 26,1791, thesonof Benj amnBabbage, aLondonbanker . Char l eswas a somewhat si ckl y youthwhose educati onwas i r r egul ar andmainl y conducted at t he hands of pr i vat e tutors . As a youthhewashi s owni nst r uctor i n al gebr a, of whi chhewas passi onatel y fond, andwaswel l -read i n t he cont i nent al mathemati cs of hi s day, par t i cul ar l yt he cal cul us of Lei bni zUponenteri ng Tr i ni t y Col l ege, Cambri dge, i n 1811, Babbagefoundhimel f i nmathemati cs f ar i n advanceof hi s t ut or s who, al ong

wthmost Engl i shmathemati ci ans, were s tu l t i f i ed by an over st r i ctadherence t o t he unfortunatenot at i ons of the cal cul usofNewonandto geometri cal form of argument i n general . As anundergraduate,wth J ohn Herschel , Peacock, and others, Babbage founded t heAnal yti cal Society f or promoti ng conti nental mathemati cs-the"D i smof Lei bni z i n opposi ti onto t he Dot' ageof theUni versi ty " I ntime t h i s campai gnwas successf ul andpl ayedan important r o l e i nt he r evi t a l i zat i on of Engl i shmathemati cs i n t he md-nineteenthcenturyI n hi s twenti es Babbage worked as amathemati ci an Hewas


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f ference andAnal yti cal Eng nes 61

el ectedaFel l owof t heRoyal Soci et y i n 1816andpl ayed apromnentpar t i n t he f ormat i on of t he Astronomcal Soci et y of London ( l a t e rt heRoyal Ast r onom cal Soci et y) i n 1820 I t wasabout t h i s t i me thatBabbage f i r s t acqui r ed t he i nterest i n cal cul at i ng machi nery t h a tbecame hi s consumngpassi on f or t he remai nder of hi s l i f e . Fromt h i s ti mehedi dnomore ser i ous mathemati cal work

Throughout hi s l i f e Babbageworked i n many i nt el l ectual f i el dsandmade cont r i but i ons that woul dhaveassuredhi s famei r r espect i veof t he D f ference and Anal yt i cal Eng nes . H s i nt er est s arewel l - r epr esent ed by hi s publ i shed works He wroteAComparati veV ewof t he Vari ous I nst i t ut i ons f or the Assurance of Li ves ( 1826)concer ni ng t he actuar i al pr i nci pl es under l yi ng l i f e i nsur ance H sTabl eofLogari thmsof t heNatural Numbersf rom1 t o 108, 000( 1827)was a paradi gmof accuracy and was ext ensi vel y used i nt o t het went i et h century . Ref l ect i ons on t he Decl i ne of Sci ence i n Eng and( 1830) i s t he best knownof Babbage' smany pol em cs agai nst t hesci ent i f i c i nst i tut i ons of hi s dayandf uel ed much debat e at t he t i meandafter. Babbage' s i nterest i n t h i s area i s al so seen i n hi s i mpor t antr ol e i n t he establ i shment of t he Associ at i on f or t heAdvancement ofSci ence ( adi r ect outgrowthof t hepubl i cat i on of Decl i ne of Sci ence)andt he Stat i st i cal Soci et y ( l a t e r t heRoyal Stat i sti cal Soci et y) , andi nhi s extensi ve cont act s wi t h cont i nent al sci ent i f i c i nst i tut i ons . Ont heEconomy ofMachi neryandManufactures ( 1832) , t he best knownofBabbage' sbooks, i s amast er l y st udyof t hemanufacturi ngt echni quesof hi s dayand t hei r economc base. I t i s seen by some as l ayi ng t hefoundati ons of the study of oper at i ons research The NnthBri dgewater Tr eat i se ( 1837) i s t hemost cur i ous of Babbage' sworksI t waswr i t t en as Babbage' s unsol i ci t edaddendumt o t heBri dgewaterTr eat i ses, whi ch ai med to prove t he exi st ence of Godthrough t her i chness of nat ur al phenomena By anal ogy wth hi s machi ne,Babbagepost ul at ed t he exi stence of a hi er ar chy of nat ur al l aws ( ani dea t hat rose t o promnence i n t he t went i et h cent ur y wth t hedevel opment of r e l a t i v i s t i c andquantummechani cs) andused t h i si dea t o pr ovi de a r at i onal expl anat i on of nat ural mr acl es . Theaut obi ogr aphi cal Passagesf romt he L i f e of aPhi l osopher ( 1864) i sa charmngthough i di osyncrat i c vi ewof ni net eent h- cent ur y l i f e . I tcont ai ns t he most extensi ve account s i n Babbage' s hand of t hepr i nci pl es and capabi l i ti es of hi s machi nes t hough, unf or t unat el y,wr i t t en at avery el ement ary l evel .Despi t e hi s many achi evements, t he fai l ure t o construct hi scal cul at i ngmachi nesand, i n part i cul ar, t he f ai l ure of t hegovernment


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t o suppor t hi s work ( a s we s ha l l descr i be l a t er ) , l e f t Babbage i n h i sdecl i ni ng year s a di sappoi nt ed and embi t t ered mn Hi s b i t t e r , bu twel l - warr ant ed, campai gn agai nst s t r e e t musi ci ans becam an e a s yc l i c h e of t he l a s t years of hi s l i f e and t he basi s of t he " i r a s c i b l e ge ni u s "mtht h a t so poor l y represents t he strengths of hi s per sonal i t y and h i sst upendous achi evement s . Babbage di ed at hi s home on DorsetS t r e e t , London, onOct ober 18, 187 1 .

TheGenesi s of theDi f ferenceEngi ne

Te i dea of an aut omat i c cal cul at i ng machi ne f i r s t came t oBabbageabout 1820 . I n one account , wri t t enmny years l a t e r ,he descr i bes howhewasengagedwi t h hi s f r i e n d , t he ast r onomer J ohnHerschel , i n pr oof r eadi ng a s e t of t a b l e s prepared f o r astronomcalcal cul at i ons . I n amoment of exasper at i onwi t h t he e r r o r s they f o u nd,Babbage r emar ked, " I wi sh t o God these cal cul at i ons had beenexecut ed by steam " Her schel ' s r e p l y , " I t i s qui te possi bl e, " s e tBabbage t hi nki ng and i n a shor t t i me, a f ewdays, he had f ormul atedt he general i dea of t he machi ne that l a t e r becam known as t heDi f f er ence Engi ne .

The i dea of t hemthodof di f f er ences, whi ch under l i es Babbage' sf i r s t aut omat i c cal cul at i ng machi ne, wasmuch i n vogue a t t ha t t i meConsi der t he f ormul aT=x +x+41

of t he vari abl e x . I t gener at es a sequence of val ues f or T-many o fwhi chhappen t o be pr i me number s, as seen i n t he t a b l e i n Fi gur e 2 1 ,I f wetake t he di f f er ences bet ween successi ve val ues of T , t he col umnl abel ed 0 i n t he t a b l e , these so- cal l ed f i r s t di f f er ences f o l l o w q u i t e asi mpl e ru le . I f wetake t he di f f er ences bet ween t he di f f er ences, knowna s t he seconddi f f er ences, t he r e s u l t i s evenmores t r i ki ng- t he seconddi f f er ence, i s a const ant . Wth t h i s knowl edge, t he t a b l e can b eb u i l t up i n a very si mpl e way, a s shownby t he box i n t he t a b l e . Taket he second di f f er ence, 2 , and add i t t o t he f i r s t di f f er ence t o f o rmanewf i r s t di f f er ence4+2=6


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Take t h i s new i r s t di f f erence andadd i t t o t he t abul ar val ue t o f ormanewt abul ar val ue47+6=53

By si mpl y r epeat i ng t h i s process t he t abl e of t he f unc t i onTmaybeextended i ndef i ni t el y usi ng no ot her mathemati cal oper at i on t hansi mpl e addi t i on ( Fi gur e 21) .

gur e2 1 . Tabul at i onof aquadr at i c usi ng t hemethodof di f f er ences .The box shows t he successi ve updati ng of t he di f f er encesr equi r ed t o f ormone newt abul ar val ue

The process can be gener al i zed . I n our exampl e t he seconddi f f er ence i s const ant because t he f unct i on T s a quadr at i c, i . e . , apol ynomal of degree 2 I f t he f unct i on Twere acubi c, such as

t he second di f f er ence woul d var y, but t he t h i r d di f f er ence, t hedi f f er ence between successi ve second di f f er ences, woul d beconst ant . I n gener al apol ynomal of degr ee nwi l l haveaconstant nt hdi f f er ence and each successi ve newval ue of t he f unct i on can beobt ai nedby nsi mpl e addi t i ons .


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The usef ul ness of di f f er ence t echni ques i s gr eat l y i ncr eased byt he f a c t t hat any secti on of a wel l - behaved cont i nuous f unct i on c anbe appr oxi mat ed by a pol ynom al . The shorter t he secti on and t hehi gher t he degr ee of t he pol ynom al t he cl oser t he appr oxi mat i on . Soi f wew shed t o t abul ate a f unct i on, such as as i ne or t he t i me of s u n s e t ,i t i s onl y necessary t o di vi de t he f unct i on i n t o short enough i n t e r v a l sand f i n d a s u i t a b l e appr oxi mat i ng pol ynom al f or each i nt e r v al .( Mat hemat i cal t echni ques f o r doi ng t h i s wer e much i mpr oved l a t e ri n t he ni net eent h century . ) Themethodof di f f er ences can thenbeusedt o t abul ate t he f unct i on t hr oughout t he i n t e r v al . Thi s pr ocess i s knowna s sub- t abul at i on .Babbage r eal i zed t hat a machi ne coul d car r y out t h i ssub- t abul ati on process . F i r s t , he needed amechani sm o r s t o r i n g ,s e pa r a t e l y , t he numbers cor r espondi ng t o t he val ues of t he t a bu l a rv a l u e, T , t he f i r s t d i f f e r en ce , t he second di f f er ence, et c . and amechani smt o add each di f f er ence t o t he val ue of t he pr ecedi ngdi f f er ence . Aquadr ati c, f o r exampl e, coul d be t abul ated by t hemachi ne shown schemat i cal l y i n Fi gur e 2 . 2 .

Fi gur e 2. 2 . Schemat i c ar r angement of t he oper at i ons r equi r ed t ot abul ate a quadr at i c us i ng t he method of di f f er ences . Themechani smof Babbage' s Di f f er ence Engi ne cor r esponds exact l yt o t h i s schemat i c .

By ear l y 1822 Babbage had constructed j u s t such a machi ne ast h i s and appl i ed i t t o t he t abul at i on of

amongot her f unc t i ons - t he f i r s t t h i r t y val ues bei ng t abul ated i n t woand a hal f mnutes . Unf or t unat el y, Babbage' s f i r s t Di f f er ence Engi nehas not sur vi ved and hi s not es and dr aw ngs a r e l o s t . Theonl y det a i l swehave of i t ar e cont ai ned i n several s h o r t l e t t e r s hewr ot e i n 1822 .I t was pr obabl y s im l a r i n general ar r angement anddesi gn t o t he l a t e r


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f ference andAnal yti cal Engines 65

desi gn of 1830 . I t i s i mpor t ant t o not e t h a t t h i s model was awor ki ngmachi ne, t hough undoubt edl y very l i m t e d i n i t s numer i cal capaci t yI deas f o r amoregeneral range of cal cul at i ng machi nes, f o r extr act i ngroots of equat i ons, mul t i pl yi ng, andcomput i ng pr i mes, had occurredt o Babbage at t h i s t i me and had been p a r t i a l l y worked o u t , but nod e t a i l s r emai n .Acompl et e Di f f er ence Engi ne r equi r es, i n addi t i on t o t hemechani sms shown i n Fi gur e 2. 1 , ameans f o r cont r ol l i ng t he acti onsof t he var i ous p a r t s so t h a t t hey a r e per f or med i n t hecorrect sequence .I t seems t h a t Babbage' s or i gi nal model act ed aut omat i cal l y i n t h i sway, but we have no i dea of t hemechani sms empl oyed

Fromt he begi nni ng, Babbage was concer ned w t h produci ngaccurate mat hemat i cal , ast r onom cal , and ot her t a bl e s . Per f or m ngt he necessar y cal cul at i ons i s onl y hal f of t he pr obl em Theot her hal fi s t o t r a ns f e r t h e cal cul at ed r e s u l t s t o t he pr i nt ed page, whi ch, i f donemanual l y and w t h t hemovabl e t ype of t he day, i s anot her great sour ceof error . Babbage, t h e r e f o r e , pr oposed t h a t t he Di f f er ence Engi neshoul d bemade t o prepare mechani cal l y t he type or pl a t e s needed f o rpri nt i ng Nomechani smf o r t h i s was i ncl uded i n t he f i r s t model butBabbage was car r yi ng out i ndependent exper i ment s w t h suchmechani sms at t he same t i me .

he Proj ect to Bui l d theDfferenceEngi neBabbage r ecogni zed hi s model of 1822 t o be j ust t h a t , a modelfromwhi ch a f i na l pr oduct i on machi ne coul d be devel oped,gi ven t h e s ubs t a nt i a l resources i n t i me, e f f o r t , andmoneyt hatwouldbe r equi r ed I n t h a t t he manner of gover nment suppor t f or researchand devel opment w t h whi ch we a r e nowf am l i a r di d not exi st i nBabbage' s day, hecommencedbycommuni cat i ngnews of h i s modeland t he p o s s i b i l i t i e s i t opened t o hi mt o t he s c i e n t i f i c communi t y,most not abl y i n an open l e t t e r t o Si r HumphreyDavy, pr esi dent oft he Royal Soci et y.Babbage' s achi evement s br ought himmmedi at e accl ai mHewasawar ded t he f i r s t Gol dMedal of t he Ast r onom cal Soci et y of LondonSomeof t h i s accl ai mmust have beenduet o h i s i ngenui t y i n r educi ngt he ment al task not j ust of ar i t hmet i c but of an ext ended sequence ofar i t hmet i c operat i ons t o a mechani cal mechani smMost l y, however ,


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Computi ng Bef ore Comput er s 66

i t was due t o t he per cei ved i mpor t ance of t he Di f f er ence Engi ne i nt he pr epar at i on of mat hemat i cal t a bl es .I n t he ear l y ni net eent h century t ab l es wer e t he onl y commona i dt o cal cul at i on- t he Thomsde Col mr cal cul at i ngmachi newas j u s ts t a r t i n g pr oduct i on i n 1822, and s l i d e r u l es , w t h t h ei r l im tedaccuracy, wer e r a r e l y f ound out si de such s pec i a l i s t appl i cat i ons ast he cal cul at i onof exci se dut y on s p i r i t s . Anymeans t oeconom ze t hepr oduct i on and, es pec i a l l y , ensur e t he accur acy of t a bl es wasofmaj ori mpor t ance . Nowher ewas t h i s so evi dent as i n t he prepar at i on of t heastronomcal t ab l e s t o ai d navi gat i on a t sea These t ab l es had t o ber ecomput ed annual l y, and t he consequences of e r r o r s i n t a bl es ,t r ansl ated i n t o e r r o r s i n navi gat i on at sea , coul d be most ser i ous .Ther ef ore, Babbage' s proj ect was of maj or i mpor t ance t o a n at i on ,such as Br i t a i n , t ha t r el i ed f or much of i t s weal t h onover seas trade .Babbage' s l e t t e r t o Si r Humphr ey Davy, and t he evi denti mpor t ance of t he Di f f er enceEngi neas assessedby t he s c i e n t i f i c mnof hi s day, l ed t o suppor t fromt he Br i t i sh gover nment t owar ds t hedevel opment of theDi f f er enceEngi ne f o r t heprepar at i on of pr a c t i c a lt abl es . The grant, i n i t i a l l y f i f teen hundred pounds but r i s i ngevent ual l y t o ar ound seventeen t housand pounds, was never cl earl yf ormul ated t o embody t he comm tment s and expect at i ons of e i t herBabbageor the gover nment . I t was pr obabl y seen by t he gover nmentas an ex grati a grant- i n- ai d t o Babbage, a gr ant w t hout commtmntor expectat i on, but was c er t a i n l y seen by Babbage as a commtmntt o t he const r ucti on proj ect by t he gover nment .Thi s l ack of a f ormal ar r angement l ed t o d i f f i c u l t i e s ; Babbageconsi der ed t h a t t he gover nment r eneged on i t s agr eement . Thegover nment gave no further suppor t t o t he const r uct i on a f t e r 1833whenBabbage' s rel at i onshi p w t h t he engi neer J oseph Cl ement , whowas bui l di ng t he Engi ne, r eached an i mpasse. I t i s pr obabl y notcoi nci dent al that 1832 mrked t he passage of t he Gr eat Ref ormB i l land t he f i r s t ext ensi on of t he vot i ng f r anchi se i n Bri tai n Fr omhatt i me, gover nment pat r onage, of t he s o r t that had suppor t edBabbage,was no l onger p o l i t i c a l l y vi abl e, t hough i t was not un t i l 1842that t het ermnat i on of gover nment suppor t was mde e x p l i c i t t o BabbageO Babbage' s rel at i onshi p w t h Cl ement we have l e s s di r e c tevi dence . The const r ucti on of t he Di f f er ence Engi ne was a verydemandi ng pi ece of preci s i on engi neer i ng f or i t s day, t hough t heexi st i ng port i ons of t he cal cul at i ng mchani sma r e proof t h a t t henecessary pr eci si on coul d be obtai ned by t he devel opment ofappr opr i at e and speci al i zed t o o l s and s k i l l i n t h ei r appl i cat i on . I t


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seems, however, t hat preci si on was carr i ed t o extremes andappl i edi n areas, such as t he decorat i ve f i n i sh of support col umns, t o whi chi t was i r r el evant . I t seems al so t hat C ement hadgraspedt he pot ent i alfor prof i t of an open-ended government j ob and Babbage fel texpl oi tedby th is .Babbage, on t he other hand, treated C ement as hi s servant andseems not t o have grasped t hat i n the decade that the D f ferenceEng ne was bei ngbui l t i t haddecl i nedf rombei ngthemaj or part ofCement' s work t o one j ob amongmanyof a successf ul eng neeri ngworkshop I t i s not surpri si ng then t hat Babbage' s demands t hatC ement rel ocat e hi s workshop to bet t er sui t theD f ference Eng neproj ect were countered by huge f i nanci al demands Thi s i mpassestoppedconst r uct i on workon t he D f ferenceEng ne i n March 1833,and i t was never resumedThe demands for preci si on i n t he manufacture of t he D f ferenceEng ne had a maj or i nf l uence i n t he devel opment of t he Br i t i shmachi ne t ool i ndustry J oseph Whi tworth, the l eadi ngmachi ne-toolmaker i n themd-ni neteenthcent ury, hadbeenempl oyedbyC ementon t he D f ference Eng ne work Whi tworth' s devel opments ofstandardi zedscrewt hreads, f or exampl e, are t r aceabl e t o C ement' swork i n t he same di rect i on f or t he D f ference Eng ne There seemsmuch t ruth i n the observati on that "Babbage made C ement, andC ement madeWhi tworth. " Late i n Babbage' s l i f e t he governmentrecei ved evi dence f rompromnent eng neers t hat the i nvestment i nthe D f ference Eng ne had been ampl y repai d by i t s spi n- of f i ntoBr i t i sh i ndust r y .

I t i s agreat pi ty t hat whenworkon theD f ferenceEng neceasedi t was cl ose t o compl eti on Henry Babbage l at er esti matedt hat onl ya f ur t her f i v e hundredpounds woul dhave suf f i ced . Babbage coul dreadi l y have f oundthe funds however hi s f eel i ngs and at t i t udes t oboth the government and C ement coul d not at the t i me havecountenanced hi s doi ngso I ndeed, these f eel i ngs di dmucht o f ormhi s embi tteredat t i t udes as an ol derman

Wthi n a year or two, Babbage' s mndhadmoveda l ongwaytowards the muchmore compl ex and i nt el l ect ual l y rewardi ngAnal yti cal Eng ne. There was then nowayhe woul dhave returnedt o t heori gi nal D f ferenceEng nedesignandbrought i t t ocompleti on,evenhadeventsmadet hat f easi bl e .


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gur e 2. 3 . El evat i on andpl an dr aw ngs of Babbage' s Di f f er enceEngi ne as pl anned about 1830 . The cal cul at i ng mechani sms ont he l e f t ; t he axes of f i g u r e wheel s f o r t he t abul ar val ue ( f a r r i ght )and s i x di f f er ences a r e c l ea r l y v i s i b l e . Thepr i nt i ngmechani sm sont he r i ght , and t he movi ngt a b l e car r yi ng t he stereot ype pr i nt i ngpl a t e and t h e s ec t o r car r yi ng t he di gi t - type punches a r e v i s i b l e i nt he center of both dr aw ngs .he Desi gn of the D f ference Engi neThe Di f f er ence Engi ne consi st ed of two maj or par t s - t hecal cul at i ngmechani smandt hepr i nt i ng and control mechani smThese a r e cl ear l y seen i n F i gur e 2. 3 , whi ch shows t he gener alar r angement of t he Di f f er ence Engi ne as pl anned about 1830 Apor t i on of t he cal cul at i ng mechani smwas assembl ed i n 1832 t odemonst r at e t o a commt t ee of t he Royal Soci ety t hat t he pr oj ect waspr oceedi ng s a t i s f a c t o r i l y . That por t i on, shown i n F i gur e 2. 4 , i s aboutone- t hi r d of t he hei ght and one- hal f of t he w dt h, or aboutone- sevent h of t he ent i r e cal cul at i ng mechani smAl most a l l of t he

parts of t he ent i r e cal cul at i ng mechanismhad been made, but notassembl ed, whenworkon t he pr oj ect st opped ear l y i n 1833 .

gur e 2 4 . The por t i on of t he c a l c ul a t i ng mechani sm of t heBabbage' s Di f f erence Engi ne assembl ed i n 1832 Recor ds ofnear l y a hundr ed f uncti ons t abul at ed by Babbage w t h t h i spor t i on have sur vi ved


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Di gi t s ar e represented i n t h e Di f f er ence Engi ne by t he r o t at i ona lpos i t i on of hori zont al gear wheel s . Anumber i s madeupof a s e r i e sof these f i g u r e wheel s rotat i ng about a common v e r t i c a l axi s . Thebot t ommos t wheel represents u n i t s , t he next t e n s , t he next hundr eds,and so on (Auser can i magi ne a deci mal poi nt l ocat ed bet ween anypai r of f i gure wheel s pr ovi ded t h i s i s done consi st ent l y t hr oughoutt he Di f f er ence Engi ne. ) Babbage used t he t ermAxi s t o mean a s t a c kof f i g u r e wheel s t h a t together s t o r e a number as a col l ect i on ofdeci mal d i g i t s . Thee n t i r e Di f f er ence Engi ne consi st s of an ax i s f o rt he t abul ar val ue of t he f unct i on, anot her a x i s f or t he di f f er ence, at h i r d axi s f o r t he second di f f er ence, and so on f or as manyorders ofdi f f er ences as a r e desi r ed These axes stand besi de one anot her , asshown i n Fi gur e 2. 3 , w t h t he axi s of t he t abul ar val ue nearest t o t hepr i nt i ngmechani smTheDi f f er ence Engi ne i s b u i l t onqui t e a l ar ge s c a l e , w t h f i g u r ewheel s about 6 i nches ( 15 cent i meters) i n di amet er spaced v e r t i c a l l yabout 3 i nches ( 7 . 5 cent i met er s) apar t on t he axes . ( These wheel s ar ebehi nd t he numberedwheel s v i s i b l e i n Fi gur e 2 . 4 . ) No cal cul at i ngmachi ne bef ore or si nce Babbage has used such l arge component s .The l arge scal e i n Babbage' s desi gns i s poss i bl y t r aceabl e t oant i c i pat ed gover nment expect ati ons based on t he pr oport i onscommon i n naval equi pment . The l ar ge s i z e pr obabl y di d l i t t l e t os i mpl i f y t he at t ai nment of t he des i r ed accuracy of machi ned p a r t swhi l e addi ng consi der abl y t o t he cost and manuf act ur i ng d i f f i c u l t i e s .Eachaxi s served not j u s t as a number s t o r e but al so as an addi ngmechani smAddi t i on occur r ed i n two steps t h a t w l l be expl ai nedw t h reference t o addi ng t he f i r s t di f f er ence t o t he t abul ar val ue .I ns i de each f i r s t di f f er ence f i g u r e wheel there i s amechani smhati s rotated t hr ough j us t as many s t e p s as t he val ue stored by t he f i gurewheel . I f t he u n i t s f i g u r e wheel stands a t 3 , t he mechani smw l l movet hr ough three s t e p s . Thi s mot i on i s conveyed by gear i ng t o t hecor r espondi ng f i gure wheel of t he t abul ar val ue axi s . I f t he l a t t e r stoodat 5 i n i t i a l l y , i t w l l be movedthree s t e p s t o stand a t 8 . Thi s processoccur s si mul t aneousl y i n t he t e n s , hundr eds, t housands, and otherd i g i t posi t i ons .

I t mayhappen t h a t addi t i on t o a f i gure wheel w l l generat e a carrythat must be pr opagat ed or added i n t o t he next hi gher d i g i t pos i t i on .I f t he uni t s d i g i t of t he t abul ar val ue were i n i t i a l l y 8 and3 i s added,i t w l l movef orward three pl aces and comet o stand at 1 , but a carrymust al so be pr opagat ed i n t o t he tens f i gure wheel of t he t abul arval ue . Carry pr opagat i on i s compl i cat ed by t he f a c t t hat i f t he tens


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f i gure wheel al r eady st ands a t 9i t wi l l bemovedf orwardby t he carryt o stand a t 0andanewcarry wi l l be pr opagat ed i nt o t he hundredsf i gur ewheel . I n t he Di f f er enceEng nethese consecut i ve car r i es mypr opagat e, as on occasi on they must , f romhe uni t s up through t hemost s i gni f i cant f i gure wheel .Each addi t i on, t her ef ore, consi sts of two di sti nct steps-thesi mul t aneous addi t i on of a l l f i gures of t he f i r s t di f f er ence t o t hecor r espondi ng f i gures of t he t abul ar val ue, and t he consecut i vepr opagat i on of car r i es f romhe uni t s up t o t hemost si gni f i cant di gi tsas r equi r ed

Tabul at i on of a f unct i on i nvol ves t he r epet i t i on of t h i s basi caddi t i on pr ocess f or eachof t he orders of di f f er ence i nvol ved . Aseachaxi s i s al so an addi ngmchani smhe t abul at i on of acubi c f unct i onf romhi r ddi f f er ences, f or exampl e, r equi r es si x steps f or each t abul arval ue produced

1 . Addi t i on of t hi r ddi f f er ence di gi t s t o seconddi f f er ence di gi ts2 Carrypr opagat i on amongsecond di f f er ence di gi ts3 Addi t i on of seconddi f f er ence di gi t s t o f i r s t di f f er ence di gi ts4 Carrypr opagat i on amongf i r s t di f f er ence di gi t s5. Addi t i on of f i r s t di f f er ence di gi t s t o t abul ar val ue di gi ts6 Carry pr opagat i on amongt abul ar val ue di gi ts .

Thi s process i s shown schemat i cal l y i n Fi gur e 25 . Negat i venumers may be hand ed wth no addi t i onal mchani smbyr epr esent i ng thems thei r ten' s coml emnts

i gur e 25 . Tabul at i on of a cubi c, showngsequent i al updat i ng oft he di f f er ences .


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Thi s scheme i s r eadi l y extended t o hi gher or der di f f e r e nc es .Tabul ati on froms i x t h di f f e r e nc es , as pl anned f or t he Di f f er enceEngi ne, woul d r equi r e t wel ve s t e p s f or each t abul ar val ue pr oduced .Babbage f ound a way t o rearrange t he cal cul at i on so t h a t onl y f o u rsteps wer e r equi r ed f o r each t abul ar val ue pr oduced i r r e s p e c t i v e o ft he numer of di f f er ences i nvol ved. Thi s i s c h a r a c t e r i s t i c of t hesophi sti cat ed l ogi cal consi derat i ons under l yi ng Babbage' s desi gns .Babbage obser ved t h a t when t he f i r s t di f f erence i s added t o t het abul ar v al u e, i n s t e p s f i v e and s i x , both t he t h i r d di f f erence andsecond di f f er ence axes ar e i d l e . Hecoul d thus add t he t h i r d di f f er encet o t he second di f f er ence, s t e p s one and two, at t he same t i me as t hef i r s t di f f er ence i s added t o t he t abul ar val ue . Steps one and twoover l ap steps f i v e ands i x . Thus onl y f our u n i t s of t i me, f o r s t e p s t h r e et o s i x , a r e needed f o r each t abul ar val ue pr oduced . Thi s rearrangedmnner of doi ng t he cal cul at i on i s shown i n Fi gur e 2 6 . I n moder nt erm nol ogy wewou d c a l l t he ar r angement of har dwar e t o performa cal cul at i on i n t h i s wayapi pel i ne .

Fi gur e 2 6 . Tabul ati on of a cubi c, showi ng t he over l appi ng ofupdat i ng used i n t he Di f f er ence Engi ne so t h a t t he cal cul at i ngt i me i s i ndependent of t he number of di f f er ences used .

Theover l appi ng i dea can be extended t o hi gher di f f er ences anda newt abul ar val ue can al ways bepr oduced i n f our s t e p s . I n g en er a l ,al l t he even di f f er ences ar e added t o t he odddi f f er ences i n two s t e p sandal l t heodddi f f er ences a r e then added tothe even di f f er ences ( andt he f i r s t di f f er ence i s added t o t he t abul ar val ue) i n two further s t e p s .Not onl y does thi s rearranged formof t he cal cul at i on saveconsi derabl e t i me but i t al somkes t he ar r angements f o r dr i vi ng t hecal cul at i ng mechani smmuch si mpl er .


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The cal cul at i ng mchani smof t he D f ference Engi ne i s real l yqui t e strai ghtforward as demnstrated by the ear l y date at wh chBabbageproducedhi s f i r s t demnstrati onmdel . Somecoml i cati oni s added by t he (very necessary) apparatus that Babbage added t oensure t hat t he mch ne woul d cal cul at e accuratel y wth gr eatr e l i a b i l i t y. Al thoughBabbage l ater f oundmuchsiml er mchani smf or addi t i on and car r i age, t he cal cul at i ng part woul d have beenper f ect l y successf ul i f i t s constructi on had been compl eted aspl anned The success of t he porti onshown i n Figure 2. 4proves t hat .

There are amng Babbage' s papers a number of t abul at i ons ofshor t sect i ons of t he l ogari thmuncti onwththi s sml l porti onof theD fference Engi ne These show that Babbage understood howt oobtai n rounded val ues for pri nti ng wthout any addi t i onalcal cul at i on . ) Had the cal cul at i ng part been coml eted, Babbagemght wel l havedi scoveredsomnewabul at i on techniquesbecausehe always expected that there woul d be thi s ki nd of feedback onanal ysi s once an automti c cal cul at i ngmch ne was avai l abl e . Thetwenti ethcenturyprovedh mi ght .

I f the cal cul ati ng mchani smof the D f ference Engi ne i sst r ai ght f orwar d, the pr i nt i ng and control mchani smi s not . I t ssophi st i cati on and the consi derabl e i nt el l ectual e f f o r t expended byBabbage on i t s ref i nemnt di dmuch t o l ay t he foundati ons f or t heAnal yti cal Engi ne andmke i t s very rapi ddevel opmnt possi bl e I tmayal so havedel ayed t he coml eti onof t he D f ferenceEngi ne to af at al extent .

The D f ference Engi ne was i ntended t o pr i nt an ent i re page oft abl es automati cal l y f romhe i ni t i al sett i ngof the di f f er ences . Fi gure2. 7 shows asampl eof seven- f i gure l ogari thmabl es t ypi cal of thosetheD fferenceEngi newas i ntended t o prepare

gure2. 7 . Asaml eof seven- f i guremthemti cal t abl es l a i dout i namnner possi bl ewtht heD fferenceEngi ne.


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The f u l l seven- f i gur e l ogar i t hmi s pr i nt ed onl y f o r t he f i r s t e n t r yof t he l i n e . Ot her ent r i es showonl y the f our l e s s s i gn i f i c ant d i g i t s .The three mos t s i gni f i c ant d i g i t s ar e pr i nt ed onl y i f they changeddur i ng t he pr ecedi ng l i n e ( hence they appear on l i n e 1553) or i n t hecal cul at i on of t he f i r s t entry f or t he present l i n e ( l i n e 1556) . Thecol umns and r ows ar e not evenl y spaced but r a t h e r an addi t i onal gapi s l e f t , a f t e r every f i f t h col umnand f i f t h r ow t o gui de t he r e ad er ' seye . Al l of these f eat ur es coul d be obt ai ned w t h the Di f f erenceEngi ne .Each d i g i t was pr i nt ed by punchi ng a type i n t o a s o f t met alstereotype pr i nt i ng pl ate The par t i c ul ar d i g i t s of t he tabul ar f unct i onpr i n t ed , and t he i r number , wasdetermned by sel ect i ng andcount i ngwheel s i n t he pr i nt i ng mchani sm Another count i ng wheeldet erm ned t he numer of t abl e col umns and t he i r separ at i ons . Thi swasact ual l y awheel of s i x t y posi t i ons, so t h a t t en col umns coul d beobt ai ned by r epeat i ng t he control pattern s i x t i mes . By s u i t a b l ypr ogrammng t h i s wheel , s i x , t wel ve, or f i f t e e n col umns, f o r exampl e,coul d be obtai ned f o r t he pr i nt i ng of t r i gonomet r i c t ab l e s . A i m l a rwheel cont r ol l ed t he spaci ng of col umns, and i t was possi bl e to pr i ntt ab l e ent r i es by col umns i nst ead of rows i f desi r ed . The pr i nt i ng ofthe l eadi ng three d i g i t s was cont r ol l ed by a t r i p mchanismc t i v a t edby t he appr opr i at e carry pr opagat i on i n t he t abul ar val ue ax i s .

The pr i nt i ng and control mchanismunder went a maj or r edesi gnabout 1832, so t ha t i n t he f i n a l desi gn i t woul dhave been verymuchl onger than shown n Fi gur e 2 . 3 . Thi s r edesi gnmar ks amaj or advancei n Babbage' s under st andi ng of cont rol i deas . I n t h e ear l i er desi gn t hevar i ous counti ng wheel s acted d i r e c t l y t o carry out t hei r programf unct i on t hemsel ves . I n t he l a t e r desi gn t hey al ways put i n t o gear aconnect i on f r omt he mai ndr i ve t o carry out the f unct i on . Thecontrolmechani sm t h e r e f o r e , t ransmts very l i t t l e power and t he wei ght oft he mchanism o be dr i ven does not l i m t t he compl exi t y of t hecontrol appar atus . Thi s i dea was ext ended t o t he cont r ol of t hecal cul at i ngmchanismwhi ch was over l apped so f ar as pos si bl e w tht he p r i n t i n g ) , resul t i ng i n a des i gn very s im l a r t o t he bar r el s l a t e rempl oyed i n t he Anal yt i cal Engi ne .


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The Or i gi ns of t he Anal yt i cal Engi neDs p i t e t he u t i l i t y of t he Di f f er ence Engi ne as a pr ac t i c a l a i d t otabl e maki ng i t i s , i n t he mat hemat i cal s en se , an ext r emel yl i m t e d i nst r ument f o r i t i s onl y d i r e c t l y capabl e of handl i ngpol ynom al s . The power s of t he engi ne a r e i ncreased great l y i f i t i sarranged so t h a t t he hi gh- order di f f er ences can be effected by t het abul ar f unct i on or l ow- order di f f er ences . For exampl e, t he si nef unct i on s a t i s f i e s t he di f f erence equat i onI f we coul d make t he second di f f erence r e l a t e t o t h e t abul ar val uei n t h i s way, t he si ne f unct i on coul d be t abul at ed ex ac t l y , w t hout t heuse of pol ynoma l appr oxi mat i ons . Al as, t he Di f f er ence Engi necannot car ry out t he mul t i pl i cat i on r equi r edby t he above f ormul a .I n l a t e 1822, a f t e r t he compl et i on of t he f i r s t model of t heDi f f er ence Engi ne, Babbage commenced expl or i ng t he s o r t s off eedback f unct i ons t h a t coul d be cal cul at edby t he Di f f er ence Engi ne .These have d e f i n i t i o n s si m l ar t o

What mot i vat ed Babbage t o expl or e t hese f unct i ons we do notknow, but despi t e t h e i r anal yt i c i n t r a c t a b i l i t y they came t o exer ci se aconsi der abl e f asci nat i on upon h m Two br i ef account s of t hef uncti ons wer e publ i shed by Babbage i n 1822 t he Di f f er ence Engi nei ncl uded addi t i onal t r a n s f e r gear i ng t o enabl e i t t o cal cul at e suchf unct i ons, and t he model ass embl ed i n 1832 had addi t i onal f a c i l i t i e sf o r t h i s pur pose . About f i f t y of these f eedback f unct i ons, cal cul at edw t h t he model , a r e t abul at ed i n Babbage' s not ebooks, and t heconcept pr ovi ded t he basi s f or hi s ar gument s about mr acl es i n t heN nt hBr i dgewat er Treat i se .When const r ucti on of t he Di f f er ence Engi ne ceased i n 1833,Babbage returned i n earnest t o f eedback f unct i ons, such as t he s i ne,whi ch he char act er i zed s t r i k i n g l y as "the Engi ne eat i ng i t s own t a i l . "Al t hough Babbage' s exact t r a i n of t hought at t h i s st age i s unknown,i t seems t h a t he f i r s t real i zed t h a t mul t i pl i cat i on coul d be carr i ed outby repeated addi t i on i f t he a b i l i t y t o step numbers upor downon t heaxes ( mul t i pl y or di vi de by t e n ) was pr ovi ded. Di vi s i on can be


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per f ormed as repeat ed subt ract i on but i t i s a " t ent a t i v e" process, f o rweneed t o be abl e t o exam ne t he r e s u l t of one s ubt r a ct i o n , whetherposi t i ve or negat i ve, bef ore knowng what t o do next . Babbage a l s of ound a substant i al l y f a s t e r method of carry propagat i on, t heAnt i c i pat i ng Car r i age, t h a t made carry propagat i on a p a r a l l e l r a t h e rthan a sequent i al process . The compl exi t y of t h i s newmechani smforced t heabandonment of t he ar r angement s of t heDi f f erence Engi newhere each storage axi s i s al so an adder . I n t he Anal yt i c al Engi nethere i s a separate " s t o r e " f o r number s and a " m l l , " or ar i t hmet i c u n i t ,where cal cul at i ons a r e madeBabbage' s i deas devel oped very qui ckl y , ai ded by t hecommandi ng knowl edge of control mechani sms that hehad g ai n edfromt he pr i nt i ng part of t he Di f f erence Engi ne . By l at e- 1834 t hebas i c pl ans f o r t he Anal yti cal Engi ne had been f ormul at ed Them l lwas separat e f romt he s t o r e . Mul t i pl i cat i on anddi vi s i on werec a r r i e dout i n t he ml l by combi nat i ons of si mpl er operat i ons under t he

d i r e c t i o n of one or morebarr el s . Thesequence of operat i ons orderedby t he b a r r e l s i ncl uded what t odaywe c a l l " l oops" and by a l t e r n a t esequences dependent on ar i t hmet i c r esul t s t hat arose dur i ngcal cul at i ons ( t oday known as " br anchi ng" ) . The cal cul at i onper f ormed was di rect ed by "super" barr el s that i n i t i a t e d t r a n s f e r s o fnumber s bet ween t he s t o r e and ml l and s t a r t e d t he sequences o foperat i ons of t he subsi di ary b a r r e l s . The super b a r r e l s al so i ncl udedl oopi ng and branchi ng c apabi l i t i e s . ( I n md- 1836 t he super b a r r e l swere r epl aced by s t r i ngs of J acquar dpunched cards . ) Wt hi n about ayear of t he cessati on of const r uct i on of t he Di f f erence Engi ne,Babbage had f ormul at ed t he basi c desi gn of a uni versal c al c ul at i ngmachi ne . Most of t he r emai nder of hi s l i f e was spent i n ref i ni ng t hedet a i l s of t h i s desi gn.

TheAnal yti cal Engi neThere was not one desi gn of t he Anal yt i cal Engi ne, but manyNew i ns i ghts of bot h a l ogi cal and mechani cal n a t u r econt i nual l y opened up newp o s s i b i l i t i e s t o Babbage f or hi s de s i gn ,whi chwas, t h e r e f o r e , i n an al most cont i nual s t a t e of f l u x .Between 1834 and 1837 Babbage devel oped i n out l i ne formseveral possi bl e arr angement s of t he bas i c s torage and c al c ul a t i nguni t s . Someof these were of consi der abl e i nt e r e st , such as one wi t h


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twom l l s t h a t coul d be used separat el y andi n par a l l e l f o r cal cul at i onson30- di gi t number s or l i nked t oget her as a s i n g l e ml l f or cal cul at i onson60- di gi t number s, but none wer e devel opedbeyondapr el i m nar ystage . However , bywor ki ng w t h t heseearl y desi gns, Babbagegai nedt he f a c i l i t y w t h l ogi cal desi gn i deas t h a t he was l a t e r t o expl oi t s oef f ect i vel y .Many- di gi t numbers a r e c har a c t er i s t i c of Babba ge' s desi gns .Thei r use gi ves consi der abl e accur acy, and al so a l ar g edynamc rangef or number val ues . Babbage di d not consi der t he use of a f l oat i ngpoi nt number r epr esent at i on, whosecompl exi t y t he desi gner s of ear l yel ect r oni c d i g i t a l comput er s al so avoi ded .By 1837 Babbage had s e t t l e d ona str ai ght f or war d ar r angement ,w t h a s i n g l e ml l and s t o r e , t h a t i s very si ml ar t o ear l y el ect r oni cd i g i t a l comput er s . Thi s ar r angement was al t er ed l i t t l e i n t he f ol l ow ngy e a r s , but t he desi gn was r ef i ned and el abor at ed t o a consi der abl eext ent . By 1847, when t h i s desi gn work ended, t h e r e was l i t t l e doubtt h a t an Anal yti cal Engi ne coul d have been b u i l t had t he necessari l yconsi der abl e r esour ces been avai l abl e . Herewe descr i be t he desi gnas i t st ood i n t he mddl e years bet ween 1838 and 1840 . I t may beconsi der ed r epr esent at i ve of Babbage' s pl ans .Fi gur e 2. 8 shows t he gener al ar r angement of Babbage' s Pl an 16,dat ed August 1840 . The f i gur e i s act ual l y a pl an dr aw ng of howt hemechani smwoul d have appear ed f r o mabove . I t al so ser ves ver yef f ect i vel y as a l ogi cal di agr am of t he f unct i onal p a r t s of t hemechani smand t h e i r i nt er connecti on . Thi s has been emphasi zed i nt he f i gur e .On t he r i ght of Fi gur e 2 . 8 i s t he s t o r e . Thi s consi sts of f i g u r ewheel s, si m l ar t o t hose i n t he Di f f er ence Engi ne, arrangedonv e r t i c a laxes on ei t her si de of a s et of "racks" or t oot hed bars . The racksconvey number s bet ween t he s t o r e axes andt he ml l ont he l e f t of t hef i g u r e .

Numbers consi st ed of f o r t y deci mal d i g i t s . Negat i ve number s, i na si gn- and- magni t ude r epr esent at i on, were i ndi cat ed w t h a separatesi gn wheel on each s t o r e axi s . I n a mechani cal cal cul at i ng machi net he bi nar y number syst em has no especi al advant age, wher eas adeci mal not at i on uses l e s s appar at us and i s easi er f or a human t oi nt er p r e t . Babbagecar ef ul l y examnednumber bases bet ween 2 and100 at var i ous st ages i n t he desi gn . The si gn- and- magni t uder epr esent at i on i s conveni ent f or mul t i pl i cat i on and di vi si onoper at i ons, but compl i cat es addi t i on and subt r act i on .

TheAnal yti cal Engi ne uses a much si mpl er mechani sm han t he


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Fi gure 2 . 8 . Babbage' s Pl an 16 f o r t he Anal yti cal Engi ne i n August1840 . The o r i gi n al pl an dr aw ng of t he mechani smhas beenannot at ed t o show t he f unct i onal rel at i onshi ps of t he p r i n ci pa lp a r t s .


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Di f f er ence Engi ne t o s t o r e and read out numbers . Toread a numbereach f i gure wheel i s turned backwar ds t o st and a t i t s 0 posi t i on . I nt he process t he wheel w l l r o t a t e t hr ough j u s t a s manyd i g i t posi t i onsa s t he d i g i t i t ori gi nal l y s t o r e d , and t h i s mot i on i s conveyed vi a ther acks t o t he ml l . Number r eadout i s therefore des t r u c t i v e, but t henumber read can be restored t o t he s t o r e a xi s , i f desi r ed, by l eavi ngt he s t o r e i n gear w th t he racks as t hey a r e returned t o t h e i r s t a r t i n gposi t i on a t t he end of t he number t r ansf er . I n t he ml l t he destr uct i ver eadout has more compl ex e f f e c t s and most number axes compr i se adoubl e s et of f i gure wheel s t h a t are used al t er nat el y- one s etr ecei vi ng what t he ot her s et gi ves of f .The capaci t y of t he s t o r e i s uncl ear fromt he f i gur e , as t he r acksmay e a s i l y be ext ended f ur t her to the r i ght . Babbage spoke a t var i oust i mes of fromone hundr ed t o one t housand number s i n t he s t o r e .Because t he Anal yt i cal Engi ne woul d have been f a i r l y sl ow onehundr ed number s woul d pr obabl y have suf f i ced f o r a l l pr a c t i c a lpurposes . I n t h a t case t he r acks woul d have been about 10 f eet ( 3met er s) l ong .Babbage al so pr oposed t o have appar at us t o read numbers f romandpunchnumber s t o J acquar d cards . Ther e qu i s i t e appar at uswoul dhave communi cat ed w t h t he store r a c k s , but t he det a i l s a r e notknown Pr i nt i ng appar at us f o r maki ng stereotype pl a t e s was al soi nt ended, but Babbage suggest ed t h a t t h i s mght be a separatemachi ne dr i ven by punched number cards .Qui t e l ong t r a i n s of gear i ng mght be i nvol ved i n t he t r a n s f e r ofnumber s fromone pl ace t o another i n t he Anal yt i cal Engi ne . Thenecessar y l ooseness and backl ash i n t he gear i ng, r equi r ed t o ensureeasy mechani cal a c t i on , mght have accumul at ed i n t hese l ongt r a n s f e r s t o such an ext ent a s t o make t he oper at i on of t he machi neuncer t ai n . Babbage overcame thi s d i f f i cu l t y by a ser i es of"l ocki ngs"- wedges t h a t come between gear t e e t h t o br i ng themaccur at el y t o t h e i r correct posi t i on . Thi s i s exact l y anal ogous t o t hepr ovi si on of ampl i f i cat i on i n el ectr oni c l ogi c gat es t o ensure t h a t t heout put vol t age l e v e l s conf or mt o a standar d s et of val ues i r r es pec t i veof t he par t i cul ar l ogi c c i r c u i t i nvol ved . Al t hough t he l ocki ngs addedmuchmechani cal compl exi t y t o t heAnal yt i cal Engi ne, Babbagewel lunder st ood t h a t they wer e es s en t i a l i f t he machi ne was t o workrel i abl y . Thi s f eat ur e i s characteri st i c of the sophi st i cat i on ofBabbage' s desi gns and gi vesmuch conf i dence t h a t t he machi ne, i fb u i l t , woul dhave workedsuccessf ul l y .On t he l e f t of Fi gur e 2 . 8 i s t he m l l , or central processi ng uni t .


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Computi ng Bef or e Computers 80

Thi s consi st s of a number of axes of f i gure wheel s arranged arounda s e t of " c e n t r a l wheel s . " The central wheel s ar e used t o t r a n s f e rnumber s w t hi n t heml l and pl ay a ro le anal ogous t o t h at of t he r a c k si n t he s t o r e . Theml l woul d have been about 6 f ee t ( 2 met er s) i ndi amet er and 15 f e e t ( 5 met er s) hi gh .The i ngress ax i s , I , and t he egress a x i s , ' A, ar e used as b u f f e r sf or number t r a n s f e r s bet ween t he s t o r e and t he ml l . The head and t a i laxes, Aand ' A , t oget her const i t ut e a doubl e- l engt h ( 8 0 - d i g i t )acc umul at or t o hol d t he pr oduct i n mul t i pl i cat i on and t he di vi dendand r emai nder i n di vi s i on . A i s also used as a s i ngl e- l engt hacc umul at or i n addi t i on andsubt r act i on . The t abl e axes, t o T9, ar eused i n mul t i pl i cat i on and di vi s i on .The axes F , ' F and " F i dent i f y t he three Ant i c i pat i ng Carrymechani sms i ncor por at ed i n t he Anal yt i cal Engi ne . As we saw t heDi f f er ence Engi ne used a sequent i al formof carry pr opagat i on . I f t hehundr eds f i gure wheel , f o r exampl e, st ood at 9 and r ecei ved a c a r r yfrom he tens f i gure wheel , i t woul dmove f or war d one d i g i t p o s i t i o nt o stand at 0and pr opagat e a carry t o t he t housands f i gure wheel . Thet housands w l l not recei ve t he carry u n t i l t he hundr eds f i g u r e wheelhas actual l y moved f or ward . I t i s t h i s del ay t h a t causes t he c a r r ypr opagat i on t o be sequent i al and t ake consi der abl e t i me .The Ant i c i pat i ng Carry i s so cal l ed because i t ant i ci pat es t h i ssequent i al act i on . The t housands may recei ve a carry f o r oneof t woreasons : e i t h e r t he hundr eds f i gure wheel movedpast 9 t o 0 dur i ngt he addi t i on step and so gener at ed a carry ; or t he hundreds f i g u r ewheel st ands at 9 and so w l l pr opagat e acar ry r ecei ved, bywhat evercause, f r omt he t ens f i gurewheel . The ant i c i pat i ng carry i ncor por at esa mechani sm t he "carry chai n, " t h a t di rec t l y i mpl ement s t hese t wol ogi cal a l t e r n at i v e s . The ant i c i pat i ng carrymechani smcan t h e r e f o r edet ermne, bef ore any f i gure wheel i s moved, whi ch ones shoul drecei ve a carry and al l of t hese can be moved f or ward t hr ough oned i g i t posi t i on si mul t aneousl y . The sequent i al and ant i c i pat i ng c a r r ymechani sms ar e contrasted i n Fi gure 2 9 .The ant i c i pat i ng carry was pr obabl y t he i dea of whi ch Babbagewas most proud . Hi s aut obi ogr aphy cont ai ns a del i ght f ul s t o r yconcerni ng i t s i nvent i on and descr i bes how hi s pr i nci paldraught sman had t hought Babbagehad t aken l eave of h i s senses f o reven cont empl at i ng i t s p o s s i b i l i t y . Per haps i t was t he base of t heenor mous conf i dence Babbage exhi bi t ed i n devel opi ng t he l og i c aldesi gn of t he Anal yt i cal Engi ne .

The carry chai n of t he ant i c i pat i ng carry woul d have demanded


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gure 2. 9 . Babbage' s two methods of car r i age propagati on i naddi ti on The upper exampl e shows the sequenti al carrymechani smused i n t heD fferenceEngi ne Correspondi ngdi gi tsare f i r s t added simul taneousl y ( uni t s t o uni t s, t ens t o tens, andsoon) and car r i ages warned, as shown by a star. The car r i agepropagati on then proceeds sequent i al l y f romt he uni t s di gi tupward The l ower exampl e shows the ant i ci pat i ngcarri ageusedi n t he Anal yt i cal Engi ne The addi t i on process i s unchanged, butal l of the car r i ages are propagatedsimul taneousl y

averyhi ghdegreeof preci si on i n i t s manufacture Butmeasurementsmadeof par t s bui l t f or t he D fferenceEngi ne showt hat the requi si t edegreeof accuracyhadbeenobtai nedon amassproducti onbasi s byabout 1830 There seem no basi s f or the common bel i ef t hatBabbage' s machi nes coul dnot havebeenmadewth t he technol ogyavai l abl e i n hi s day, thoughdoubtl ess i t woul dhavebeenexpensi ve.Rather , i t seem t hat af t er hi s bi t t er experi ence i n attempti ngt o bui l dt he D f ference Engi ne, Babbage chose t o concentrate on thei nt el l ect ual i ssues r ai sed i n the desi gn of t he Anal yti cal Engi ne andbui l t onl y smal l t r i a l model s t o ver i f y hi s desi gns Babbage' s designswerevery thoroughl ydevel opedand t hemechani cal i ssues caref ul l yconsi dered Theyweremuchmore thanj us t pen andpaper sketchesof an i dea


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TheMethodsEmployedf or Ari thmeti c Operati ons

The methods used f or mul t i pl i cat i on and di vi si on i n t heAnal yti cal Engi ne are qui te str ai ght f orward, al though t heamount of apparatus requi red i s subst ant i al and there are manyt echni cal refi nements.Mul t i pl i cat i on commences by taki ng one of the operands fromthe store and repeatedly add ng i t to i tse l f to formthe f i r s t ni nemul t i pl es, wh chare stored on the t abl e axesT1 t o T9 Themul t i pl i eri s thentakenonedi gi t at a t i me, commenci ngwth t he uni t s, andthecorrespond ngmul ti pl e i s sel ect ed from he tabl e axes andaddedt otheproduct, wh chi s accumulatedon t he headandt a i l axesAn d ' A .Themul ti pl es on the tabl e axes are al l steppedupone di gi t posi t i onandtheprocess i s repeatedwth t he next di gi t of themul t i pl i er . Thevari ous act i ons are overl appedi n suchamnner that eachwholestepr equi r es only a si ngl e addi t i on time The r esul t i s a double- lengthproduct that i s returned to the store .

Dvi si on i s s im la r . Aabl e i s f i r s t madeof theni nemul t i pl es ofthe di vi sor . The twomost si gni f i cant di gi ts of the remi nder, on theheadandt a i l axes, are comparedsiml taneouslywtht he two l eadi ngdi gi ts of eachof t he mul t i pl es t o est i mate t he next quoti ent di gi t . Thi sguessw l l ei ther becorrect or onetoo l arge . The sel ect edmul ti pl e i ssubtracted fromt he remi nder and i f t h i s becomes negati ve t hedi vi sor i s addedback t o gi veanewremi nder . Thenewremi nder i ssteppedoneplaceandt he process i s repeated I n1840Babbagefoundways of overl app ng t he acti ons so that di vi si on al so took only asi ngl e addi t i on ( subt racti on) timei rr espect i ve of whether t he quot i entdi gi t hadbeen corr ectl y guessed i ni t i al l y

Add ti on and subt racti on are muchmore compl ex processesbecauseof the si gn-and-mgni tude r epr esent at i on usedi n t he store .I n mul t i pl i cat i on and di vi s i on the si gns of t he operands can bei gnored, theoperandstreatedasunsi gned, andt he correct si gnsimplyi nsert ed i nto the r esul t . Aegati ve operand, however, turns anaddi ti on i nto a subt r act i on, and vi ce versa, and so the f uncti onperformedi n theml l must bechangedby t he si gn

Mul ti pl i cati on anddi vi si on are sl ow, taki ng one to twomnutesandone t o fourmnutes r espect i vel y, sotheoverheadtime i n f etchi ngoperands and stori ng resul ts i s not important (al though these areoverl appedwth other acti ons as f ar as possi bl e) . I n addi t i on andsubt racti on the fetchi ngandstor i ng, i nclud ng t he conversi ons from


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and t o t he si gn- and- magni t ude r epresent at i on, takemuch l onger thant he two t o three seconds r equi r ed by t he oper at i on i t s e l f .Babbage avoi ded these d i f f i c u l t i e s t o a l ar ge ext ent by t aki ng ashi s basi c oper at i on the addi t i on or subt r acti on ( i n any mxture) of awhol e s et of oper ands . Any p a r t i a l sums coul dbe wr i t t en t o t he s t o r eas r equi r ed I n e f f e c t , Babbage pr ovi ded r esi dual storage i n t he ml l( on t he head axi s A) of t he par t i al sums i n a compl ement numbersystem Thi s i s exactl y t he same as t he pr acti ce i n el ect r oni c d i g i t a lcomputer s, save t h a t there t he r esi dual storage ( i n r e gi s t e r s ) i savai l abl e f o r other t ypes of oper at i ons as wel l .Babbage or gani zed addi t i on and subt r acti on so that oper andscoul d be i n d i f f e r e n t s t ag es of pr ocessi ng si mul t aneousl y . Thi s i sshown i n Fi gur e 2. 10 . Eachoper and i s f i r s t fetched f r omt he s t o r e t ot he i ngr ess axi s . The val ue i s then added or subt r act ed, as r equi r ed,f r omt he t o t a l on t he head ax i s , A Thi s t o t a l i s i n a t e n ' s compl ementr epr esent at i on. I f t he par t i a l sum s t o be wi t t en t o t he s t o r e , i t i stransferred t o t he egr ess axi s and conver t ed, i n t he process, t o asi gn- and- magni t ude r epr esent at i on . F i n a l l y , t he r e s ul t i s wr i t t en t ot he s t or e . The control i s ver y i ngeni ous l y ar r anged so t h a t t hemaxi mumpossi bl e t hr oughput i s achi eved- t he l im ta t i on bei ngaccess t o t he s t o r e v i a t he r acks t o fetch oper ands or s t o r e r e s u l t s . Thi swas a st upendous achi evement i n l ogi cal desi gn .

TheAnal yt i cal Engi ne pr ovi ded other ar i t hmet i c oper at i ons, butt he compl et e s et i s not c l ea r because Babbage di d not l i s t what wewoul d now c al l t he user i n s t r u c t i o n s et . Ther e wer e var i ant s ofmul t i pl i cat i on and di vi si on f o r use when onl y a l im t ed number ofd i g i t s of t he r e s ul t wer e r equi r ed These wer e used, f o r exampl e, i nt he ear l y s t e p s of f i ndi ng a squar e root by i t e r a t i v e f or mul as . I n e a r l i e rdesi gns t he squar e r o ot operat i on had been i mpl ement ed as anel ement ar y oper at i on. Dur i ng t he s l ow mul t i pl i cat i on and di vi s i onoper at i ons t he i ngr ess ax i s , I , and t he ant i ci pat i ng c a r r y , " F, coul d beused d i r e c t l y w t h t he s t o r e as a di f f er ence engi ne . Possi bl y t h i s wasi nt ended f o r cal cul at i ng si mpl e pol ynom al f unct i ons r equi r ed by t hemai n c al c ul a t i on , or sub- tabul at i on of f unct i ons bet ween pi votalval ues cal cul at ed i n t he m l l . I t i s a ni ce exampl e of t he use off unct i onal par al l el i sm


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Fi gure 2. 10 . The ar r angement of addi ti on i n t he Anal yti cal Engi ne .The f our s t e p s , whi ch may be over l apped w th one anot her f ord i f f e r e n t operands, ar e descr i bed i n t he t e x t .


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D fference andAnal yti cal Engines 85

TheControl Mechani smTe al gor i t hms used i n t he Anal yti cal Engi ne, whi l e si mpl eenough i n out l i ne, ar e compl exwhenexamned i n det ai l - f orthere ar e a l ar ge number of har dwar e component s t h a t must workt oget her . Babbage achi eved control of t h i s machi ner y w t h ahi er ar chi cal systemof mechani sms .At t he l owest l ev el t he control i s exer ci sedby " b a r r e l s " ( Fi gur e2. 1 1 ) s im l a r t o those empl oyed i n musi c boxes, b a r r e l organs, andmany aut omat a f am l i a r i n Babbage' s day . St uds may be scr ewed t ot he surface of t he ba r r e l i n any desi r ed pattern . When t he b a r r e ladvances, bymovi ng i t s ax i s si deways, one v er t i c al rowof st uds a c t sby pr essi ng agai ns t cont r ol l evers i n a pattern det erm ned by t hear r angement of t he st uds . The l ev er s i n turn engage and di sengaget he t r ansm ssi on of power f r om t hemai n dr i ve shaf t t o t he var i ousmechani sms of t he Anal yti cal Engi ne. One " v e r t i c a l , " or l i n e of s t u d s ,det erm nes t he act i ons dur i ng one addi ng cyc l e . I n pr a ct i c e, a b a r r e lhad f r om50 t o 100 v e r t i c a l s , each w t h as many as 200 st uds .

gur e 2 . 11 . Aa r r e l mechani smused i n t he Anal yti cal Engi ne . Theb a r r e l may det erm ne t he sequenci ng bet ween i t s own v e r t i c a l sby both t he uncondi t i onal and condi t i onal mechani sms shownher e .


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Someof t he l ever s sel ect ed by t he bar r e l cont r ol i t s r ot at i on f r omone v er t i c al t o anot her . Some of these ac t uncondi t i onal l y . Ot her sestabl i sh a path by whi ch acondi t i onal event i n t he Anal yt i cal Engi ne,such as a car ry pr opagat i on from hemost s i gn i f i c ant d i g i t pos i t i onof a number ( a r unni ng u p ) , can move t he b a r r e l t o anot her ver t i c al .Thus, each v er t i c al can det er m ne whi ch v er t i c al w l l succeed i t andwhi ch condi t i onal event s ar e t o ef f ec t t he choi ce . I nt e r es t i ngl y ,Babbage' s ar r angement s pr ovi de onl y f or what t odaywec a l l " r e l at i veaddr ess i ng"- i . e . , a v er t i c al can spec i f y howf ar t o go t o t he nextver t i c a l t hat i s t o ac t but cannot spec i f y i t s absol ut e l ocat i on . Babbagehadnoconcept equi val ent t o t he moderni dea of t he address of awordAv er t i c al may speci f y a re turn t o a pr evi ous v er t i c al and i n t h i smanner what wenowc a l l " l oops" are pr ovi ded .Babbage' s use of bar r e l s was muchmore el abor at e t han t h i s .Ther e wer e, i n general , several bar r el s - Fi gur e 2 . 8 shows three but

somedesi gns had as many as seven . I f these al l turn t o get h er , t hee f f e c t i s not hi ngmore t han di vi di ng up an i nconveni ent l y l arge b a r r e li n t o a gr oup of smal l er ones . But i n t he case of addi t i on andsubt r act i on t he bar r el s cont rol l i ng t he r e gi s t e r s I , A, and "A andassoc i a ted mechani sm acted i ndependent l y of one anot her ,r espondi ng i n part t o l oc al condi t i onal event s, yet cooper at i ngt oget her t o i mpl ement t he s t r i n g of addi t i on and subt r act i onoper at i ons . Thewhol e ar r angement i s enor mous l y sophi st i cat ed yetf i n e l y j udged t o best expl oi t t he c apabi l i t i e s of t he c al c ul at i ngmec hani sms .

The bar r el s spec i f y i n det a i l howmul t i pl i cat i on, di v i s i on ,add i t i on , s ubt r a c t i on , and ot her ar i t hmet i c oper at i ons , ar e t o becar r i ed out . Theuser of t he Anal yt i cal Engi ne woul d regard t h e s eari t hmet i c operat i ons as basi c and s peci f y a cal cul at i on i n t er ms o fthemas el ement ar y f unct i ons . We have, therefore, a hi er a r c hi c alar r angement of t he cont r ol .For t he hi gher l evel of c ont r o l , Babbage, i n 1836, adopt ed t hepunched cards devel oped by J acquard f o r pat t er n- weavi ng l ooms andused extens i vel y s i nce 1810 ( Fi gur e 2. 12) . Acard i s pressed agai nstt he ends of cont rol l ev er s so t h a t t he pattern of hol es i n t he carddetermnes whi ch l ev e r s ac t . The act i on i s ent i r el y anal ogous t o t hestuds compr i si ng a v er t i c al ona b a r r e l . TheJ acquar d cards are s t r u n gt oget her by narrowr i bbons so t hat t hey compr i se, i n essence, a papertape . I t i s pos s i bl e , by mec hani sms s im l a r t o t hose used t o r o t a t e t hebar r el s , t o move f or war d or backwar ds t hr ough a s t r i n g of t heJ acquard cards . I n e f f ec t t he s t r i n g of J acquar d cards i s equi val ent t o


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i gur e 2. 12 . Aacquard pattern-weavi ng l oom cont r ol l ed bypunched cards . Babbage adoptedandgeneral i zed th is mchani smf or the user - l evel speci f i cati on of cal cul at i ons f or the Anal yt i calEngi ne Courtesy Sci ence Museum London

a barrel w than i ndef i ni tel y l arge numer of vert i cal s . The adopti onof the cards represents l ess a conceptual breakthrough than apragmti c i mrovemnt on theearl i er useof asuper barrel t o speci f ythe steps of a cal cul at i on . Babbagemade l i t t l e of t h i s developmntand i t s i mortance has been considerably over- romnti ci zed by t heanalogyw thmodern uses of punchedcards .

rogrammng the Analyti cal Engi neAthoughBabbage' smchani cal technology i s vastl y di f f erentf rommodern el ectr oni cs, i t i s rel at i vel y easy t o f i ndanal ogi es

that make hi s organi zati on of the cal cul at i ng uni ts and st or age andthei r control by t he barr el s and J acquard cards f am l i ar t o moderncomuter users . I t i s onl ywhenwecome t o examne the fac i l i t iesavai l abl e f or programmng t he Anal yti cal Engi ne that Babbage' sdesi gns begi n t o l ook strange t o modern eyesTwo str i ngs of J acquard cards were needed to speci fy a


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cal cul at i on t o beper f or medby t he Anal yt i cal Engi ne . Ones t r i n g , t he"operat i on cards, " speci f i ed t he ari t hmet i c oper at i ons t o beper f ormed Thesecond s t r i n g , t he "var i abl e cards, " speci f i ed t he axesi n t he s t o r e that cont ai ned t he operands andwer e t o recei ve t he r e s u l t s .These two s t r i n g s cannot be regarded as separat e parts of a s i n g l ei n s t r u c t i o n , as ar e t he operat i on andoperand f i e l d s of an i n s t r u c t i o ni n an el ectr oni c d i g i t a l comput er , because t he operat i on and var i abl ecards were i nt ended t o moveand l oop i ndependent l y of one anot herunder t he di rect i on of separat e control mechani sms .Babbage seems t o have been l ed t o separat e t he operat i on andvar i abl e cards on l argel y phi l osophi cal grounds s t emm ng fromhi sb e l i e f i n t he need t o di st i ngui sh symbol s f o r operat i on fromhose f orquant i t y i n mat hemat i cal not at i ons . These vi ews wer e probabl yrei nf orced when he consi dered t he cards necessary f o r c a l c u l a t i o n ssuch as t he sol ut i on of si mul t aneous equat i ons . There t he pattern ofoperat i ons requi red f o r carry i ng out rowreduct i ons i s very si mpl eand a st rai ght f orward l oop of operat i on cards i s readi l y found . Nosuch si mpl e l oop s t r uc t u r e ex i s t s f o r t he var i abl e c a r d s , whi ch canonl y spec i fy s i ngl e axes i n t he s t o r e . Thel oop structures t h a t wenowr ecogni ze concern r ows of t he mat r i x of c o e f f i c i e n t s of t he equat i onsand s im l a r concept s rel at ed t o t he st ructuri ng of t he dat a . As Babbagedi d not have t he concept of a var i abl e address i n t he s t o r e , n ei t he rwas t he Anal yt i cal Engi ne abl e t o cal cul at e t he l ocat i on of an operandi n t he s t o r e ; there was no way i n whi ch t he user programs coul dexpl oi t t h i s hi gher l e vel s t r u c t u r e i n t he dat a .I n r e a l i t y , weknow i t t l e of Babbage' s programmng i deas . Therei s not hi ng i n t he survi vi ng paper s i n whi ch t h i s aspect of t he machi nei s t horoughl y di scuss ed, e . g . , no th i ng correspondi ng t o aspec i f i cat i on of a user i ns t ruc t i on s et . Thi s i s t he morer emar kabl e f o ri t i s t he onl y aspect of t he desi gn t h a t i s di scussed at l engt h i n acont empor ar y paper . I n 1840, Babbagevi s i tedTuri n i n I t a l y andgavea s er i es of semnars on t he Anal yt i cal Engi ne . An account of t h e s e ,by Menabrea, was t ransl at ed i n t o Engl i sh by Ada Lovel ace, whoappended ext ensi ve not es prepared under Babbage' s c l ose gui dance .These deal w t h t he f am l i ar modern i deas of f l o w of control i npr ograms, par t i c u l a r l y t he f ormul at i on of si mpl e l oops and nest edl oops cont rol l ed by count ers . However , t hepaper andnot es c a r e f u l l yand del i berat el y s k i r t around any di scussi on of d e t a i l s of t he meansby whi ch t hese are t o be i mpl ement ed .AdaLovel ace has somet i mes been accl ai med as t he "worl d' s f i r s tprogr ammer " on t he strength of her aut horshi p of t he not es t o t heMenabrea paper . Thi s romant i cal l y appeal i ng i mage i s w t hout


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f oundat i on . Al l but one of t he progr ams c i t e d i n her notes hadbeenpreparedbyBabbagef romhree t o seven years e a r l i e r . Theexcept i onwas prepared by Babbage f o r he r , al t hough she di d detect a "bug" i ni t . Not onl y i s there no evi dence t hat AdaLovel ace ever prepared aprogramf or t he Anal yt i cal Engi ne but her correspondence w t hBabbage shows t h a t she di d not have t he knowedge t o do so .Babbage seems t o have del i berat el y empl oyed i ndependent personst o convey knowl edge of t he Anal yt i cal Engi ne t o t he w der publ i c i nexact l y t he same manner as , a decade e a r l i e r , he had used t hewel l - known popul ar i zer of sci ence Di onysi us Lardner t o convey i n t op r i n t a det ai l ed account of t he purpose of t he Di f f erence Engi ne .The concl usi on seems i nescapabl e t hat Babbage di d not have af i r mcommandof t he i s s ues rai sed by t he user- l evel programmngoft he Anal yt i cal Engi ne . I t woul dbe qui t ewrong t o i nf e r t ha t Babbagedi d not underst and programmngper s e . Themcr opr ogr ammng oft hebarr el s f o r mul t i pl i cat i on anddi vi si on showcommandof t he basi cbranchi ng and l oopi ng i deas and hi s s k i l l s i n t hemcr opr ogr ammngof addi t i on and subt ract i on showcompl et e vi rt uosi t y . I t was f romhi sbase t h a t Babbageexpl ored t he i deas of user- l evel programmng Thei ssues of dat a st r uct ur i ng si mpl y di d not ar i se at t hemcr opr ogr ammng l ev el . There i s some evi dence t o suggest t h a tBabbage' s i deas wer e movi ng i n t he di recti ons now f aml i ar i nconnect i on w t h the control mechani sms f or l oop count i ng i nus e r - l ev el progr ams . Had an Ana l y t i c a l Engi ne ever been brought t oworki ng o r d e r , there can be no doubt t h a t Babbage' s pr ogrammngi deas woul d have been devel oped g r e a t l y .Babbage real i zed t h a t the Anal yt i cal Engi ne was a uni versalcal cul at i ngmachi ne i n t he sense t h a t , gi ven s u f f i c i e n t t i me, i t coul dcarry out any possi bl e ari t hmet i c cal cul at i on . The ar gument , cl ear l ypresent ed i n si mpl e t erms i n hi s aut obi ography, i s based on threeobservat i ons . F i r s t , ar i t hmet i c operat i ons on number s of more thanf o r t y di g i t s can al ways be carr i ed t hr ough by breaki ng them nto40- di gi t segment s, so t he l i m t ed number of d i g i t s on any s t o r e axi si s no f undament al l i m t . Second, cal cul at i ons can be speci f i ed bys t r i n g s of operat i on andvari abl e cards of unl i mt ed e x t e n t , so therei s no l im t a t i o n t o t he s i z e or compl exi t y of progr ams . Thi rd, number sfromhe s t o r e can bepunched onto number cards and l a t e r read back,andt h i s provi des a backi ng s t o r e of unl i mt ed ext ent t oovercome t hel im t e d number of axes i n t he s t o r e . Thi s sophi st i cat ed ar gument hasa very t went i et h- cent ury f l avor . Babbage was not awar e t ha t t he r emght be uncomput abl e numbers, a concept t h a t der i ves f r om t heb r i l l i a n t work of Al an Tur i ng i n t he 1930s .


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Babbage' s Later Cal cul ati ngEngi nesWr k on t he desi gn of t he Anal yt i cal Engi ne ended i n 1847 . Att h a t t i me Babbage t ur ned t o t he desi gn of a Di f f er ence Engi neNo 2, expl oi t i ng t he i mpr oved andsi mpl i f i ed ari t hmet i c mechani smsdevel oped fo r theAnal yt i cal Engi ne . Thel ogi cal desi gnwast he sameas f o r t he e a r l i e r Di f f er ence Engi ne, but he empl oyed si mpl ermechani sm f o r stori ng and addi ng numbers and carry propagat i on .Thepr i nt i ngmchanismwas si mpl i f i ed so that a whol enumber wasi mpr essed on a pr i nt i ng pl ate as a si ngl e act i on rather than i n adi gi t - by- di gi tmanner . Aconvent i onal p r i n t copy, us i ng i nked r o l l e r s ,wasmade si mul t aneousl y . The control was arranged by a s i n g l eb a r r e l i n a very st r ai ght f orwardmanner . Thedesi gn and a compl et es et of draw ngs was prepared bym d- 1848 . These Babbageofferedt o t he B r i t i s h government , apparent l y t o s a t i s f y acommtmnt he f e l texi st ed i n consequence of t he f a i l u r e of the proj ect t o bui l d t he f i r s tDi f f er ence Engi ne. The gover nment showedno i n t er es t i n t he newdesi gn .Babbage appears t o have done no more work on c a l c u l a t i n gengi nes u n t i l t he Scheut z Di f f er ence Engi ne ( descri bed i n t he nextsect i on) was br ought t o London i n 1855 . To t he surpri se of some,Babbagebecam an a c t i v e andvi gorous pr omot er of t he Scheut zesand t h ei r machi ne .I ns pi r e d, perhaps, by t he Scheut zes' success, Babbage returnedt o desi gnworkon t he Anal yti cal Engi ne i n about 1856or 1857, whenhewas65 years ol d . I n t h i s newphase of workBabbagewas a c t i v e l yi nt er est ed i n bui l di ng an Anal yti cal Engi ne w t h hi s ownresources .The l o g i c a l desi gnwas somwhat si mpl i f i ed b u t , most i mpor t ant l y,f ar si mpl er and cheaper met hods were proposed t o i mpl ement t hebasi c mechani sm Babbage f i r s t exper i ment ed w t h sheet met alst ampi ng and pr essi ng f o r maki ng gear wheel s and s im la r p a r t s .L a t e r , he adopt ed pressure di e casti ng f or makingpart s- a new yi nvent ed t echni que t h a t di d not see ext ensi ve commer ci al use unt i lt he end of t he ni net eent h cent ur y . Babbage b u i l t manyexper i ment almodel s of mechani sm usi ng thesenewt echni ques, and, at t he t i meof hi s deat h i n 1871, amode of a si mpl eml l andpr i nt i ngmchanismwas near compl et i on . ( Fi gur e 2. 1 3 )Thi s l a s t work of Babbage i s poor l y under st ood because of t hedi sor gani zed and chaot i c nat ure of t he materi al s t h a t r emai n . Thei mpressi on i s unavoi dabl e t hat i n t h i s l a t e r workBabbagehadl o s t t hef i n e t ouch of geni us exhi bi t ed i n hi s e a r l i e r wor k, al t hough hi s vari ous


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gur e 2. 13 . Amodel of t he m l l of t he Anal yt i cal Engi ne that wasunder const r uct i on a t t he t i me of Babbage' s deat h . Thehor i zont alracks communi cat e number s bet ween t he twonumber axes i n t hecent er andt o t he pr i nt i ngmechani sm t t he r i ght . An ant i ci pat i ngcar r i age mechani sm s l ocat ed bet ween t he number axes . Thecal cul at i ng mechani smempl oys pressure di e- cast met alcomponent s . Courtesy Sci ence Museum London

exper i ment al model s s t i l l showmuch evi dence of an i ngeni ous andenqui r i ngmnd I n f a i r n e s s , wemust note t h a t most of t h i s l a t e r workwas car r i ed on whenBabbagewas bet ween sevent y andei ght y yearsol d . Babbage' s cal cul at i ng machi nes and r el at ed mat er i al s wer ei nher i t edby hi s youngest son, Maj or - Gener al HenryPBabbage, whohad shown a st r ong i nt er es t i n hi s f a t h e r ' s work Henry Babbagedeci ded not t o cont i nue w t h t he desi gn of an Anal yt i c Engi ne buti nst ead t o devel op a manual l y operated machi ne f o r addi t i on,s ubt r a c t i o n, mul t i pl i c a t i on , and di vi s i on ( a f our - f unct i on c a l c u l a t o r ) ,i ncor por at i ng t he mechani sms pl anned f o r t h e m l l of t he Anal yt i cal


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Engi ne . Al t hough eventual l y compl eted, whenHenrywashi msel f anol dman, t h i s machi ne appear s never t o haveworked r e l i a b l y . I n anyc a s e , by t h e s t a r t of t h e twent i et h cent ur y i t had been r ender ed ar c h ai cby other devel opment s of mechani cal cal cul ati ng machi nes, so t h a tnow t stands onl y as a s c i e n t i f i c cur i o .

TheScheutz Di f ference Engi neThe f i r s t successf ul aut omat i c cal cul ati ngmachi newasdevel opedi nSweden i n t h e 1840s byGeorgScheutz and h i s son Edvar d .Thei r machi newas a Di f f er enceEngi ne based d i r e c t l y onBabbage' sdesi gn, whi ch t hey l earned about whenGeor ge Scheutz t r a n s l a t e dLar dner ' s a r t i c l e i n t o Swedi sh . I n that Lar dner ' s a r t i c l e contai ns onl yt he most general descr i pt i ons of t h e mchanismof t he Di f f er enceEngi ne ( wi t hout dr awi ngs of t h e mechani sms) i t i s a smal l s ur p r i s et h a t t he Scheut z machi ne ( Fi gur e 2. 1 4 ) l ooks very di f f e r e nt f r omBabbage' s ( Fi gur e 2. 4) .

Fi gur e 2. 14 Thecopy of t h e Scheutz Di f f er enceEngi ne b u i l t f o r t h eGener al Regi st er Of f i ce, London The p i l l a r at t he r i gh t movesacross t h e f r ont of t he f i gur e wheel s t o e f f e c t the car r i agepr opagat i on . The pr i nti ngmchanisms behi nd t he cal cul ati ngwheel s at t he l e f t . Cour t esy Sci ence Museum London


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f f erence andAnal yt i cal Engines 93

I n t he Scheut z Di f f er ence Engi ne t he f i gure wheel s, as i nBabbage' s desi gn, a r e hori zont al wheel s r o t a t i n g about a ver t i c al axi s .However , t he f i gure wheel s f o r t he t abul ar val ue ar e ar r anged i n arowona hori zont al s h e l f , whi l e t he f i r s t - and hi gher - order di f f erencesa r e ar r anged on successi ve shel ves bel ow There a r e f i v e shel ves,al l owi ng up t o fourth-order di f f er ences, and f i f t e e n d i g i t s i n al lnumbers . Asequent i al car ry pr opagat i on i s pr ovi ded by " p i l l a r s " t h a tt r a v e l t he l ength of t he mechani sm i n f r ont of and behi nd t hecal cul at i ngwheel s . As i n Babbage' s machi ne, t he odddi f f er ences areupdat ed si mul t aneousl y, then t he even di f f er ences ( and t he t abul arval ue) a r e updat ed t oget her .The pri nt i ng mechani smpunches stereotype p l at es , al l d i g i t s ofa number bei ng i mpr essed si mul t aneousl y a f t e r t he manner ofBabbage' s l a t e r desi gns . The Scheut z Di f f er ence Engi ne pr epar esonl y a s i n g l e col umn of t abul ar val ues . The page l ayout i s made upmanual l y fromthese s t r i p s . There i s no at t empt at an aut omat i cmechani smf or t h i s pur pose .Const r uct i on of t he Scheut z Di f f er ence Engi ne was compl et ed i nOct ober 1853 wi t h some assi st ance fromt he Swedi sh gover nment .I t was l a t e r t aken t o Londonwher e i t , and GeorgandEdvard Scheut z,wer e champi oned by Babbage The machi ne was exhi bi t ed i n t hePari s Exhi bi t i on of 1855, awarded a gol d medal , and wi del yaccl ai med . The or i gi na l machi ne was sol d t o t heDudl eyObser vat or yi n Al bany, NewYork, i n 1856 t o be used i n pr epar i ng ast r onomcalt a bl es . Acopy of t he machi ne was made byBryanDonki n&Co f ort he Gener al Regi st er Of f i ce i n Engl and about 1858 .Three s e t s of t a bl es wer e publ i shed t h a t had been cal cul at ed, atl eas t i n p a r t , by t he Scheut z Di f f er ence Engi ne . Aset of Speci menTabl es, i ncl udi ng a t a b l e of f i ve- f i gure l ogar i t hms of t he i nt eger s from1 t o 1000, was pr epared on t he or i gi na l machi ne i n 1856 That samey e a r , a s et of Mount ai n Baromet er Tabl es, f or assessi ng hei ght s onmount ai ns or dept hs i n mnes fromsi mul t aneous observat i ons ofbar omet r i c pressure and t emperatur e, wer e prepared by t he Engl i shcopy of t he Scheut z Di f f erence Engi ne . The maj or pr oduct i on wast he pr epar at i on of t he Engl i sh L i f e Tabl es, publ i shed i n 1864, at t heGener al Regi st er Of f i ce .Nei t her model of t he Scheut z Di f f er ence Engi ne was f ound t o bevery s a t i s f a c t o r y and t he use of bot h was qui ckl y abandoned . Anumber of f a c t o r s cont r i but ed t o t h i s f a i l u r e . The cal cul at i ons weresl ow, l argel y because of t he awkwar dcarry pr opagat i on mechani smused . The machi ne depended on f r i c t i o n al one t o keep t he f i gur e


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ComputingBefore Computers 94

wheel s i n t h e i r correct posi t i on when a number was stored-therewereno spr i ng det ent s or ot her mechani sms equi val ent t o Babbage' sl ocki ngs t o r e t a i n t he f i gur e wheel s i n pl ace . W t hout such pr ovi si oni t seems t he f i gur e wheel s e a s i l y becamedi spl aced f r o mt h e i r c o r r e c tposi t i on and t he cal cul at i on spoi l ed Al t hough t h i s coul d be det ect edby exam ni ng t he l a s t val ue pr i nt ed i t was no doubt a sour c e ofconsi der abl e annoyance. Thepr i nt i ngmechani smseems al so t o havebeen unr el i abl e and e r r o r s of t h a t s o r t woul dhave been d i f f i c u l t t odet ect . I n gener al , bot h mac hi nes were found t o be d e l i c a t ei nst r ument s t h a t r equi r ed consi der abl e s k i l l t o mani pul at e and hencewer e i l l - s u i t e d t o r out i ne use .

Later D fferenceandAnal yti cal Engi nesThat t he Scheut z Di f f er ence Engi ne possessed f a u l t s i s scar cel ysur pr i s i ng i n vi ewof i t s bei ng t he f i r s t compl et ed machi ne of i t st ype I t i s r egr et t abl e t h a t more exper i ence hadnot been gai ned w t ht he o r i gi na l mac hi ne bef or e t he Engl i sh copywas made, when t heoppor t uni t y mght have been t aken t o el i mnat e t he d i f f i c u l t i e s .Pr oduct i on of a r e l i a b l e di f f er ence engi ne r equi r ed t he i nvest ment ofnew, i nvent i ve e f f o r t t o bui l d on t he Babbage and Scheut zachi evement s . However , t h e r e wer e onl y twomoredevel opment s i nt hat di r ect i on i n t he ni net eent h centuryI n Sweden, Mar t i n Wberghadb u i l t a di f f er ence engi ne by 1860and used i t t o prepare a s e t of i n t e r e s t t a bl es f or publ i cat i on . TheWberg Di f f er ence Engi ne was both smal l er and si mpl er t han t heScheut z, t hough i t possessed t he same ar i t hmet i c capabi l i t y Themachi ne appear s t o have worked r e l i a b l y and was used i n t hepr epar at i on of l ogar i t hm c and t r i gonomet r i c t a b l e s t h a t appear ed i n1875 .I n Amer i ca, George Gr ant devel oped a smal l model of adi f f er ence engi ne i n 1871 andexhi bi t ed a compl et e mac hi ne i n 1876 .But t h i s mac hi ne soon f aded i n t o obscur i t y and appear s not t o havebeen put t o any pr ac t i c al use .I n t he t went i et h cent ur y t he use of di f f er ence engi nes i n t a b l emaki ng agai n r ecei ved somepromnence . I n t h i s case, however , t heconstr uct i on of speci al pur pose mac hi nes was not at t empt ed but wayswer e f ound t o adapt t he gener al pur pose cal cul at i ng machi nes t henon t he mar ket t o t h i s speci al pur pose . Thebest knownwork i s t h a t of


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L J . Comi e at t he Br i t i shNaut i cal Al manac Of f i ce i n t he 1920s and1930s, usi ngmul t i pl e r egi st er account i ngmachi nesmanufacturedbyBurroughs andNat i onal Cash Regi st er .

I f l i t t l e effort was made t o devel op di f f er ence engi nes, i t i sscar cel y sur pr i s i ng t hat not hi ng subst ant i al f ol l owed i n the t r adi t i onof Babbage' s Anal yt i cal Engi ne

I n I rel and an anal yt i cal engi ne was desi gned by PercyLudgateabout 1905 I n i t i a l l y t hi s work was i ndependent of Babbage' s butl at er Ludgate came t o knowandbe i nf l uenced byBabbage' s i deas .The desi gn, whi chwas pur el y mechani cal , cont ai ns some s t r i k i ngfeatures . The mechani sm i n whi ch numbers were st or ed werephys i cal l y transported f rom hememorywhen t henumber was readApseudol ogar i t hm c r epr esent at i on of di gi t s was used t o si mpl i f ybot h mul t i pl i cat i on and di vi s i on oper at i ons . Amost i nt erest i ngfeature was t he abandonment of Babbage' s separ at e oper at i on andvar i abl e cards andt he adopt i on of cont r ol by apaper t ape i n whi cheachi ns t r uct i on compri sedanoper at i on codeandf our addr ess f i el ds .Very l i t t l e i nf or mat i on onLudgat e' s desi gn has sur vi ved, and t her ei s noevi dence t hat he ever at t empt ed t o const r uct t hemachi neVery i nt erest i ngdesi gns of anal yt i cal engi nes weremade i n Spai n

i n t he 1910s and1920s byLeonardo Torres yQuevedo Torres wasa wel l -known engineer who vi gorousl y expl oi ted t he newel ectromagneti c t echnol ogi es i n the devel opment of cont r olmechani sm . He i s par t i cul ar l y wel l -known f or two f ul l y aut omat i cchess pl ayi ng automataf or t heendi ngof ki ngandrook agai nst aki ngI n 1920Tor r es const r uct ed anel ect r omagnet i c cal cul at i ngmachi net hat was dr i ven by oper ands typedon a t ypewr i t er anddel i ver ed i t sr esul t s usi ng t he same devi ce . Tor r es' s i deas f or an anal yt i cal engi newere suf f i ci ent l y wel l devel oped t hat there i s no doubt t hat asuccess f ul machi necoul dhave been bui l t i n t he 1920s had t heneedf or such amachi nebeenpr essi ng .

Ihe Importance of Babbage' s Cal cul ati ng Engi nesn t he desi gns of t heDi f f er enceEngi neandt he Anal yt i cal Engi neBabbagemade t he f i r s t maj or i nt el l ect ual cont r i but i ons towardst hedevel opment of aut omat i c di gi t al computi ngmachi nes al t houghhi s i deas were not r eal i zed unt i l over a cent ur y l at er . Twomaj orquest i ons remainabout Babbage' sworkWywere hi smachi nes not


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ComputingBefore Computers 96

successf ul l y constructed? Andwhat i nf l uence di d hi s i deas have ont he subsequent devel opment of aut omat i c comput er s?The pr esent evi dence suggests, qui t e s t r ongl y , t hat bot h t heDi f f er ence Engi ne and t he Anal yt i cal Engi ne coul d have been b u i l t

successf ul l y w t h t he mechani cal t echnol ogy a t Babbage' s di s p os a l .The cal cul at i ng par t of t he Di f f er ence Engi ne came c l ose t ocompl et i on and t he por t i on i n t he Sci enceMuseum London, wor kssuperbl y . The f a i l u r e of t h a t pr oj ect seems t r aceabl e i n part t oBabbage' s r e l a t i o n s h i p w t h t he Br i t i sh gover nment over t he f undi ngof t he pr oj ect but especi al l y t o Babbage' s rel at i onshi p w t h t heengi neer Cl ement . The l ar ge physi cal scal e of t he machi ne and ,p a r t i c u l a r l y , t he very hi gh degr ee of pr ec i s i on at t ai ned i n t hemanuf act ur e of i t s p a r t s and t he concomt ant expense seemo havebeen t he root causes of t he f a i l u r e t o br i ng i t t o compl et i on .TheDi f f er ence Engi ne has a di r e c t l i n e of descendant s throught he Scheut z t o t he Wberg andGr ant di f f erence engi nes . That t h e s ewere not ext ensi vel y used or devel oped, despi t e t he appar entcompl et e success of t heWberg machi ne, i ndi cat es t hat t he e nt i r e i de awas not wel l j udged . The sub- t abul at i on t a s k , t hough l abor i ous, wasnot t he domnant mathemat i cal task i n t he pr epar at i on of t a b l e s no r ,w t h adequat e or gani zat i on andmanagement , was i t of over whel m ngpr act i cal i mpor t ance . Babbage' s ar gument f or t he accuracy i nt ypeset t i ng made poss i bl e by machi nes ( l a t e r st rongl y hel d byHoward Ai ken) was not w del y accept ed, and Babbage' s ownl ogar i t hmt a b l e s a r e pr oof of t he accuracy t h a t coul d be obt ai ned bymanual t echni ques . Wenmachi ne sub- t abul at i on was adopt ed byComi e, i t was i n t he cont ext of a l arge- scal e mechani zat i on of t a b l emaki ng i n whi ch t he bal ance of e f f or t i n t he whol e pr oj ect was notmuchchanged .Al t hough t he Anal yti cal Engi ne coul d have been b u i l t , Babbagechose, f or most of hi s l i f e , not t o at t empt t o do so . Thi s i s a nat uralresponse t o hi s exper i ences w t h t he Di f f er ence Engi ne and t heenor mous i n t e l l e c t u a l appeal of t he quest i ons rai sedby t he Anal yt i calEngi ne . O great r e g r e t i s t he f a c t t ha t Babbagenever publ i shed adet ai l ed account of any of hi s many i deas and mechani sms . TheMenabrea-Lovel ace paper del i ber at el y concentrates on t hemathemat i cal pr i nci pl es embodi ed i n t he machi ne and compl et el yavoi ds descr i bi ng t h e i r mechani zat i on .Wt hout a det ai l ed

Babbage Difference Analytical Engine

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