,,! ,,ed9burgh ,,review1 ,,july1 #ahcd4 ,no4 ,,cxx4 ,,! ,,calculat+ ,,eng9e ,,by ,,*>les ,,babbage ,art ,i4--#a4 ,lett] to ,sir ,humphry ,davy1 ,b>t4 ,p4,r4,s41 on ! application ( ,ma*9]y to ,calculate & ,pr9t ,ma!matical ,tables4 ,by ,,*>les ,,babbage1 ,esq4 ,f4,r4,s4 #dto4 ,pr9t$ by ord] ( ! ,h\se ( ,commons4 #b4 ,on ! ,application ( ,ma*9]y to ! ,calculation ( ,a/ronomical & ,ma!matical ,tables4 ,by ,,*>les ,,babbage1 ,esq4 ,memoirs ,a/ron4 ,soc4 ,vol4 ,i4 ,"p #b4 ,london3 #ahbb4 #c4 ,address to ! ,a/ronomical ,society1 by ,h5ry ,?omas ,colebrooke1 ,esq4 ,f4,r4,s4 ,presid5t1 on pres5t+ ! fir/ gold m$al ( ! ,society to ,*>les ,babbage1 ,esq4 = ! inv5tion ( ! ,calculat+ ,eng9e4 ,memoirs ,a/ron4 ,soc4 ,vol4 ,i4 ,"p #b4 ,london3 #ahbb4 #d4 ,on ! det]m9ation ( ! ,g5]al ,t]m ( a new ,class ( ,inf9ite ,s]ies4 ,by ,,*>les ,,babbage1 ,esq4 ,transactions ,camb4 ,phil4 ,soc4 ,cambridge3 #ahbd4 #e4 ,on ,errors common to _m ,tables ( ,log>i?ms4 ,by ,,*>les ,,babbage1 ,esq4 ,memoirs ,a/ron4 ,soc4 ,london3 #ahbg4 #f4 ,on a ,me?od ( ,express+ by ,signs ! ,action ( ,ma*9]y4 ,by ,,*>les ,,babbage1 ,esq4 ,phil4 ,trans4 ,london3 #ahbf4 #g4 ,report by ! ,committee appo9t$ by ! ,c\ncil ( ! ,royal ,society to consid] ! subject ref]r$ to 9 a ,communication receiv$ by !m f ! ,treasury1 respect+ ,mr ,babbage's ,calculat+ ,eng9e1 & to report !reupon4 ,london3 #ahbi4 ,,!re is no position 9 society m enviable ?an t ( ! few :o unite a mod]ate indep5d5ce ) high intellectual qualities4 ,lib]at$ f ! necessity ( seek+ _! support by a pr(ession1 !y are unfett]$ by its re/ra9ts1 & are enabl$ to direct ! p[]s ( _! m9ds1 & to conc5trate _! intellectual en]gies on ^? objects exclusively to :i* !y feel t _! p[]s may be appli$ ) ! greate/ advantage to ! community1 & ) ! mo/ la/+ reputation to !mselves4 ,on ! o!r h&1 _! middle /ation & limit$ income rescue !m f ^? allurem5ts to frivolity & dissipation1 to :i* rank & weal? ev] expose _! possessors4 ,plac$ 9 su* fav\rable circum/ances1 ,mr ,babbage select$ sci5ce z ! field ( 8 ambition2 & 8 ma!matical rese>*es h conf]r$ on him a high reputation1 :]ev] ! exact sci5ces are /udi$ & appreciat$4 ,! suffrages ( ! ma!matical _w h be5 ratifi$ 9 8 own c\ntry1 :]e he has be5 elect$ to ! ,lucasian ,pr(essor%ip 9 8 own ,univ]sity--a *air1 :i*1 ?\gh ( inconsid]able emolum5t1 is "o on :i* ,newton has conf]r$ ev]la/+ celebrity4 ,b x has be5 ! =tune ( ?is ma!matician to surr\nd himself ) fame ( ano!r & m popul> k9d1 & :i* r>ely falls to ! lot ( ^? :o devote _! lives to ! cultivation ( ! ab/ract sci5ces4 ,?is di/9ction he owes to ! ann\ncem5t1 "s ye>s s9ce1 ( 8 celebrat$ project ( a ,calculat+ ,eng9e4 ,a proposition to r$uce ari?metic to ! dom9ion ( me*anism1--to sub/itute an automaton = a compositor1--to ?r[ ! p[]s ( ?\ght into :eel-"w c\ld n fail to awak5 ! att5tion ( ! _w4 ,to br+ ! practicability ( su* a project )9 ! compass ( popul> belief 0 n easy3 to d s by br++ x )9 ! compass ( popul> compreh5sion 0 n possible4 ,x transc5d$ ! imag9ation ( ! public 9 g5]al to conceive its possibility2 & ! s5tim5ts ( wond] ) :i* x 0 receiv$1 7 only prev5t$ f m]g+ into ^? ( incr$ulity1 by ! fai? repos$ 9 ! high atta9m5ts ( its projector4 ,?is extraord9>y und]tak+ 01 h[ev]1 view$ 9 a v diff]5t light by ! small section ( ! community1 :o1 be+ suffici5tly v]s$ 9 ma!matics1 7 acqua9t$ ) ! pr9ciple ^u :i* x 0 f\nd$4 ,by ref]5ce to t pr9ciple1 !y p]ceiv$ at a glance ! practicability ( ! project2 & be+ enabl$ by ! nature ( _! atta9m5ts & pursuits to appreciate ! immeasurable importance ( its results1 !y reg>d$ ! inv5tion ) a proportionately pr(\nd int]e/4 ,! production ( num]ical tables1 unlimit$ 9 quantity & v>iety1 re/rict$ to no p>ticul> species1 & limit$ by no p>ticul> law2--ext5d+ n m]ely to ! b\nd>ies ( exi/+ k1 b spread+ _! p[]s ov] ! undef9$ regions ( future discov]y--7 results1 ! magnitude & ! value ( :i* ! community 9 g5]al c\ld nei!r compreh5d nor appreciate4 ,9 su* a case1 ! judgm5t ( ! _w c\ld only re/ ^u ! au?ority ( ! philosophical "p ( x2 & ! fiat ( ! sci5tific community sway$ = once political c\ncils4 ,! ,briti% ,gov]nm5t1 advis$ by ! ,royal ,society1 & a committee =m$ ( ! mo/ em95t me*anicians & practical eng9e]s1 det]m9$ on con/ruct+ ! project$ me*anism at ! exp5se ( ! nation1 to be held z national prop]ty4 ,not)/&+ ! int]e/ ) :i* ?is inv5tion has be5 reg>d$ 9 e "p ( ! _w1 x has nev] yet be5 embodi$ 9 a writt51 mu* less 9 a publi%$ =m4 ,we tru/1 !re=e1 t "s cr$it w be conc$$ to u = hav+ be5 ! fir/ to make ! public acqua9t$ ) ! object1 pr9ciple1 & /ructure ( a piece ( ma*9]y1 :i*1 ?\gh at pres5t unkn[n 7except z to a few ( its probable results71 mu/1 :5 complet$1 produce important effects1 n only on ! progress ( sci5ce1 b on t ( civilisation4 ,! calculat+ ma*9]y ?us und]tak5 = ! public gratuit\sly 7s f> z ,mr ,babbage is conc]n$71 has n[ atta9$ a v advanc$ /age t[>ds completion2 & a portion ( x has be5 put toge!r1 & p]=ms v>i\s calculations2--af=d+ a practical demon/ration t ! anticipations ( ^?1 "u ^: advice ,gov]nm5t has act$1 h be5 well f\nd$4 ,!re are nev]!less _m p]sons :o1 admitt+ ! great +5uity ( ! contrivance1 h1 not)/&+1 be5 accu/om$ to reg>d x m 9 ! light ( a philosophical curiosity1 ?an an in/rum5t = purposes practically useful4 ,?is mi/ake 7?an :i* x is n possible to imag9e a great]7 has aris5 ma9ly f ! ignorance :i* prevails ( ! ext5sive utility ( ^? num]ical tables :i* x is ! purpose ( ! eng9e 9 "q to produce4 ,!re are also "s p]sons :o1 n consid]+ ! "t requisite to br+ any inv5tion ( ?is magnitude to p]fection 9 all its details1 incl9e to consid] ! delays :i* h tak5 place 9 its progress z presumptions aga9/ its practicability4 ,^! p]sons %\ld1 h[ev]1 be=e !y arrive at su* a conclusion1 reflect ^u ! "t :i* 0 necess>y to br+ to p]fection eng9es inf9itely inf]ior 9 complexity & me*anical difficulty4 ,let !m rememb] t--n to m5tion ! inv5tion ( t ma*9e--! improvem5ts alone introduc$ into ! /eam-eng9e by ! celebrat$ ,watt1 occupi$ a p]iod ( n less ?an tw5ty ye>s ( ! life ( t di/+ui%$ p]son1 & involv$ an exp5diture ( capital am\nt+ to ,l4#ej1jjj4a ,! calculat+ ma*9]y is a contrivance new ev5 9 its details4 ,its inv5tor did n take x up already imp]fectly =m$1 aft] hav+ receiv$ ! contributions ( human +5uity ex]cis$ ^u x = a c5tury or m4 ,x has n1 l almo/ all o!r great me*anical inv5tions1 be5 gradually advanc$ to its pres5t /ate ?r\gh a s]ies ( failures1 ?r\gh difficulties enc\nt]$ & ov]come by a succession ( projectors4 ,x is n an object on :i* ! light ( v>i\s m9ds has ?us be5 %$4 ,x is1 on ! contr>y1 ! production ( solit>y & individual ?\ght1--begun1 advanc$ ?r\gh ea* successive /age ( improvem5t1 & br\ght to p]fection by "o m9d4 ,yet ?is creation ( g5ius1 f its fir/ rude conception to its pres5t /ate1 has co/ little m ?an half ! "t1 & n "o-?ird ( ! exp5se1 consum$ 9 br++ ! /eam-eng9e 7previ\sly f> advanc$ 9 ! c\rse ( improvem5t7 to t /ate ( comp>ative p]fection 9 :i* x 0 left by ,watt4 ,%ort z ! p]iod ( "t has be5 :i* ! inv5tor has devot$ to ?is ent]prise1 x has1 nev]!less1 be5 demon/rat$1 to ! satisfaction ( _m sci5tific m5 ( ! fir/ em95ce1 t ! design 9 all its details1 r$uc$1 z x is1 to a sy/em ( me*anical draw+s1 is complete2 & requires only to be con/ruct$ 9 con=mity ) ^? plans1 to realize all t its inv5tor has promis$4 ,footnote #a3 ,watt comm5c$ 8 inve/igations respect+ ! /eam-eng9e 9 #agfc1 betwe5 :i* "t1 & ! ye> #aghb inclusive1 he took out sev]al pat5ts = improvem5ts 9 details4 ,bolton & ,watt _h exp5d$ ! above sum on _! improvem5ts be=e !y began to receive any return4 ,) a view to remove & correct errone\s impressions1 & at ! same "t to conv]t ! vague s5se ( wond] at :at seems incompreh5sible1 ) :i* ?is project is contemplat$ by ! public 9 g5]al1 into a m rational & edify+ s5tim5t1 x is our purpose 9 ! pres5t article4 ,fir/1 ,to %[1 ! imm5se importance ( any me?od by :i* num]ical tables1 absolutely accurate 9 e individual copy1 may be produc$ ) facility & *eapness4 ,?is we %all e/abli% by convey+ to ! read] "s notion ( ! numb] & v>iety ( tables publi%$ 9 e c\ntry ( ! _w to :i* civilisation has ext5d$1 a l>ge portion ( :i* h be5 produc$ at ! public exp5se2 by %[+ also1 t !y are nev]!less r5d]$ ineffici5t1 to a great] or less ext5t1 by ! preval5ce ( errors 9 !m2 t ^! errors p]vade n m]ely tables produc$ by individual lab\r & ent]prise1 b t !y vitiate ev5 ^? on :i* national res\rces h be5 prodigally exp5d$1 & to :i* ! highe/ ma!matical ability1 :i* ! mo/ enlight5$ nations ( ! _w c\ld comm&1 has be5 unsp>+ly & sy/ematically direct$4 ,secondly1 ,to attempt to convey to ! read] a g5]al notion ( ! ma!matical pr9ciple on :i* ! calculat+ ma*9]y is f\nd$1 & ( ! mann] 9 :i* ?is pr9ciple is br\ght into practical op]ation1 bo? 9 ! process ( calculat+ & pr9t+4 ,x w\ld be incompatible ) ! nature ( ?is review1 & inde$ impossible )\t ! aid ( num]\s plans1 sections1 & elevations1 to convey cle> & precise notions ( ! details ( ! means by :i* ! process ( reason+ is p]=m$ by inanimate matt]1 & ! arbitr>y & caprici\s evolutions ( ! f+]s ( typographical compositors are r$uc$ to a sy/em ( :eel-"w4 ,we are1 nev]!less1 n )\t hopes ( convey+1 ev5 to read]s unskill$ 9 ma!matics1 "s satisfactory notions ( a g5]al nature on ?is subject4 ,?irdly1 ,to expla9 ! actual /ate ( ! ma*9]y at ! pres5t "t2 :at progress has be5 made t[>ds its completion2 & :at are ! probable causes ( ^? delays 9 its progress1 :i* mu/ be a subject ( regret to all fri5ds ( sci5ce4 ,we %all indicate :at appe>s to u ! be/ & mo/ practicable c\rse to prev5t ! unnecess>y recurr5ce ( su* ob/ructions = ! future1 & to br+ ?is noble project to a spe$y & successful issue4 ,view+ ! inf9ite ext5t & v>iety ( ! tables :i* h be5 calculat$ & pr9t$1 f ! e>lie/ p]iods ( human civilisation to ! pres5t "t1 we feel emb>rass$ ) ! difficulties ( ! task :i* we h impos$ on ourselves2--t ( attempt+ to convey to read]s unaccu/om$ to su* speculations1 any ?+ approa*+ to an adequate idea ( !m4 ,^! tables are connect$ ) ! v>i\s sci5ces1 ) almo/ e dep>tm5t ( ! useful arts1 ) comm]ce 9 all its relations2 b above all1 ) ,a/ronomy & ,navigation4 ,s important h !y be5 consid]$1 t 9 _m in/ances l>ge sums h be5 appropriat$ by ! mo/ enlight5$ nations 9 ! production ( !m2 & yet s num]\s & insurm\ntable h be5 ! difficulties att5d+ ! atta9m5t ( ?is end1 t aft] all1 ev5 navigators1 putt+ aside e o!r dep>tm5t ( art & sci5ce1 h1 until v rec5tly1 be5 scantily & imp]fectly suppli$ ) ! tables indisp5sably necess>y to det]m9e _! position at sea4 ,! fir/ class ( tables :i* naturally pres5t !mselves1 are ^? ( ,multiplication4 ,a great v>iety ( ext5sive multiplication tables h be5 publi%$ f an e>ly p]iod 9 diff]5t c\ntries2 & especially tables ( ,p[]s1 9 :i* a numb] is multipli$ by itself successively4 ,9 ,dodson's ,calculator we f9d a table ( multiplication ext5d+ z f> z #aj times #ajjj4b ,9 #agge1 a /ill m ext5sive table 0 publi%$ to #aj times #aj1jjj4 ,! ,bo>d ( ,longitude subsequ5tly employ$ ! late ,dr ,hutton to calculate & pr9t v>i\s num]ical tables1 & among o!rs1 a multiplication table ext5d+ z f> z #ajj times #ajjj2 tables ( ! squ>es ( numb]s1 z f> z #be1djj2 tables ( cubes1 & ( ! fir/ t5 p[]s ( numb]s1 z f> z #ajj4c ,9 #ahad1 ,pr(essor ,b>l[1 ( ,woolwi*1 publi%$1 9 an octavo volume1 ! squ>es1 cubes1 squ>e roots1 cube roots1 & reciprocals ( all numb]s f #a to #aj1jjj2 a table ( ! fir/ t5 p[]s ( all numb]s f #a to #ajj1 & ( ! f\r? & fif? p[]s ( all numb]s f #ajj to #ajjj4 ,footnote #b3 ,dodson's ,calculator4 #dto4 ,london3 #agdg4 ,footnote #c3 ,hutton's ,tables ( ,products & ,p[]s4 ,folio4 ,london2 #agha4 ,tables ( ,multiplication to a /ill great] ext5t h be5 publi%$ 9 ,france4 ,9 #aghe1 0 publi%$ an octavo volume ( tables ( ! squ>es1 cubes1 squ>e roots1 & cube roots ( all numb]s f #a to #aj1jjj2 & simil> tables 7 aga9 publi%$ 9 #ahja4 ,9 #ahag1 multiplication tables 7 publi%$ 9 ,p>is by ,vois92 & simil> tables1 9 two qu>to volumes1 9 #ahbd1 by ! ,fr5* ,bo>d ( ,longitude1 ext5d+ z f> z a ?\s& times a ?\s&4 ,a table ( squ>es 0 publi%$ 9 #ahaj1 9 ,hanov]2 9 #ahab1 at ,leipzig2 9 #ahbe1 at ,b]l92 & 9 #ahbg1 at ,gh5t4 ,a table ( cubes 0 publi%$ 9 #ahbg1 at ,eis5a*2 9 ! same ye> a simil> table at ,gh5t2 & "o ( ! squ>es ( all numb]s z f> z #aj1jjj1 0 publi%$ 9 t ye>1 9 qu>to1 at ,bonn4 ,! ,prussian ,gov]nm5t has caus$ a multiplication table to be calculat$ & pr9t$1 ext5d+ z f> z #ajjj times #ajjj4 ,su* are a few ( ! tables ( ?is class :i* h be5 publi%$ 9 diff]5t c\ntries4 ,?is class ( tables may be consid]$ z purely ari?metical1 s9ce ! results :i* !y express involve no o!r relations ?an ! ari?metical dep5d5ce ( ab/ract numb]s ^u ea* o!r4 ,:5 numb]s1 h[ev]1 are tak5 9 a concrete s5se1 & are appli$ to express peculi> modes ( quantity1--su* z angul>1 l9e>1 sup]ficial1 & solid magnitudes1--a new set ( num]ical relations arise1 & a l>ge numb] ( computations are requir$4 ,to express angul> magnitude1 & ! v>i\s relations ( l9e> magnitude ) :i* x is connect$1 involves ! consid]ation ( a va/ v>iety ( ,geometrical & ,trigonometrical tables2 su* z tables ( ! natural s9es1 co-s9es1 tang5ts1 secants1 co-tang5ts1 c4 c42 tables ( arcs & angles 9 t]ms ( ! radius2 tables = ! imm$iate solution ( v>i\s cases ( triangles1 c4 ,volumes )\t numb] ( su* tables h be5 f "t to "t comput$ & publi%$4 ,x is n suffici5t1 h[ev]1 = ! purposes ( computation to tabulate ^! imm$iate trigonometrical functions4 ,_! squ>es#d & high] p[]s1 _! squ>e roots1 & o!r roots1 occur s frequ5tly1 t x has be5 f\nd exp$i5t to compute tables = !m1 z well z = ! same functions ( ab/ract numb]s4 ,footnote #d3 ,! squ>es ( ! s9es ( angles are ext5sively us$ 9 ! calculations connect$ ) ! !ory ( ! tides4 ,n aw>e t tables ( ^! squ>es exi/$1 ,b\v>d1 :o calculat$ ! tides = ,laplace1 und]w5t ! lab\r ( calculat+ ! squ>e ( ea* individual s9e 9 e case 9 :i* x occurr$4 ,! measurem5t ( l9e>1 sup]ficial1 & solid magnitudes1 9 ! v>i\s =ms & modifications 9 :i* !y are requir$ 9 ! arts1 dem&s ano!r ext5sive catalogue ( num]ical tables4 ,! surveyor1 ! ar*itect1 ! build]1 ! c>p5t]1 ! m9]1 ! gang]1 ! naval ar*itect1 ! eng9e]1 civil & milit>y1 all require ! aid ( peculi> num]ical tables1 & su* h be5 publi%$ 9 all c\ntries4 ,! increas$ exp$ition & accuracy :i* 0 introduc$ into ! art ( computation by ! inv5tion ( ,log>i?ms1 greatly enl>g$ ! numb] ( tables previ\sly necess>y4 ,to apply ! log>i?mic me?od1 x 0 n m]ely necess>y to place 9 ! h&s ( ! computi/ ext5sive tables ( ! log>i?ms ( ! natural numb]s1 b likewise to supply him ) tables 9 :i* he might f9d already calculat$ ! log>i?ms ( ^? ari?metical1 trigonometrical1 & geometrical functions ( numb]s1 :i* he has mo/ frequ5t occasion to use4 ,x w\ld be a circuit\s process1 :5 ! log>i?m ( a s9e or co-s9e ( an angle is requir$1 to ref]1 fir/ to ! table ( s9es1 or co-s9es1 & !nce to ! table ( ! log>i?ms ( natural numb]s4 ,x 0 !re=e f\nd exp$i5t to compute di/9ct tables ( ! log>i?ms ( ! s9es1 co-s9es1 tang5ts1 c41 z well z ( v>i\s o!r functions frequ5tly requir$1 su* z sums1 diff]5ces1 c4 ,great z is ! ext5t ( ! tables we h j enum]at$1 !y be> a v insignificant proportion to ^? :i* rema9 to be m5tion$4 ,! above are1 = ! mo/ "p1 g5]al 9 _! nature1 n belong+ p>ticul>ly to any sci5ce or art4 ,!re is a mu* great] v>iety ( tables1 ^: importance is no way inf]ior1 :i* are1 h[ev]1 ( a m special nature3 ,su* are1 = example1 tables ( int]e/1 disc\nt1 & ex*ange1 tables ( annuities1 & o!r tables necess>y 9 life insurances2 tables ( rates ( v>i\s k9ds necess>y 9 g5]al comm]ce4 ,b ! sci5ce 9 :i*1 above all o!rs1 ! mo/ ext5sive & accurate tables are indisp5sable1 is ,a/ronomy2 ) ! improvem5t & p]fection ( :i* is insep>ably connect$ t ( ! k9dr$ art ( ,navigation4 ,we sc>cely d>e hope to convey to ! g5]al read] any ?+ approa*+ to an adequate notion ( ! multiplicity & complexity ( ! tables necess>y = ! purposes ( ! a/ronom] & navigator4 ,we feel1 nev]!less1 t ! truly national importance :i* mu/ atta* to any p]fect & easy means ( produc+ ^? tables _c be at all e/imat$1 unless we /ate "s ( ! previ\s calculations necess>y 9 ord] to enable ! m>9] to det]m9e1 ) ! requisite c]ta9ty & precision1 ! place ( 8 %ip4 ,9 a ^w1 !n1 all ! purely ari?metical1 trigonometrical1 & log>i?mic tables already m5tion$1 are necess>y1 ei!r imm$iately or remotely1 = ?is purpose4 ,b 9 addition to ^!1 a great numb] ( tables1 exclusively a/ronomical1 are likewise indisp5sable4 ,! pr$ictions ( ! a/ronom]1 ) respect to ! positions & motions ( ! bodies ( ! firmam5t1 are ! means1 & ! only means1 :i* enable ! m>9] to prosecute 8 art4 ,by ^! he is enabl$ to discov] ! di/ance ( 8 %ip f ! ,l9e1 & ! ext5t ( 8 dep>ture f ! m]idian ( ,gre5wi*1 or f any o!r m]idian to :i* ! a/ronomical pr$ictions ref]4 ,! m num]\s1 m9ute1 & accurate ^! pr$ictions c be made1 ! great] w be ! facilities :i* c be furni%$ to ! m>9]4 ,b ! computation ( ^? tables1 9 :i* ! future position ( cele/ial objects are regi/]$1 dep5d !mselves ^u an inf9ite v>iety ( o!r tables :i* nev] rea* ! h&s ( ! m>9]4 ,x _c be said t !re is any table :atev]1 necess>y = ! a/ronom]1 :i* is unnecess>y = ! navigator4 ,! purposes ( ! m>9e ( a c\ntry ^: int]e/s are s insep>ably connect$ z ours are ) ! improvem5t ( ! art ( navigation1 w\ld be v inadequately fulfill$1 if our navigators 7 m]ely suppli$ ) ! means ( det]m9+ by ,nautical ,a/ronomy ! position ( a %ip at sea4 ,x has be5 well obs]v$ by ! ,committee ( ! ,a/ronomical ,society1 to :om ! rec5t improvem5t ( ! ,nautical ,almanac 0 confid$1 t x is n by ^? means m]ely by :i* ! seaman is enabl$ to det]m9e ! position ( 8 vessel at sea1 t ! full int5t & purpose ( :at is usually call$ ,nautical ,a/ronomy are answ]$4 ,?is object is m]ely a "p ( t compreh5sive & important subject2 & might be atta9$ by a v *eap publication1 & )\t ! aid ( exp5sive in/rum5ts4 ,a n less important & mu* m difficult "p ( nautical sci5ce has = its object to det]m9e ! precise position ( v>i\s int]e/+ & important po9ts on ! surface ( ! e>?1--su* z rem>kable headl&s1 ports1 & isl&s2 toge!r ) ! g5]al tr5d+ ( ! coa/ betwe5 well-kn[n h>b\rs4 ,x is n necess>y to po9t out "h h[ important su* k is to ! m>9]4 ,?is k1 :i* may be call$ ,nautical ,geography1 _c be obta9$ by ! me?ods ( obs]vation us$ on bo>d %ip1 b requires mu* m delicate & accurate in/rum5ts1 firmly plac$ ^u ! solid gr\nd1 besides all ! a/ronomical aid :i* c be af=d$ by ! be/ tables1 arrang$ 9 ! mo/ conv5i5t =m = imm$iate use4 ,?is 0 ,dr ,maskelyne's view ( ! subject1 & 8 op9ion has be5 confirm$ by ! repeat$ wants & dem&s ( ^? di/+ui%$ navigators :o h be5 employ$ 9 sev]al rec5t sci5tific exp$itions4#e ,footnote #e3 ,report ( ! ,committee ( ! ,a/ronomical ,society prefix$ to ! ,nautical ,almanac = #ahcd4 ,among ! tables directly necess>y = navigation1 are ^? :i* pr$ict ! position ( ! c5tre ( ! sun f h\r to h\r4 ,^! tables include ! sun's "r asc5sion & decl9ation1 daily1 at noon1 ) ! h\rly *ange 9 ^! quantities4 ,!y also include ! equation ( "t1 toge!r ) its h\rly v>iation4 ,tables ( ! moon's place = e h\r1 are likewise necess>y1 toge!r ) ! *ange ( decl9ation = e t5 m9utes4 ,! lun> me?od ( det]m9+ ! longitude dep5ds ^u tables conta9+ ! pr$ict$ di/ances ( ! moon f ! sun1 ! pr9cipal planets1 & f c]ta9 conspicu\s fix$ />s2 :i* di/ances be+ obs]v$ by ! m>9]1 he is enabl$ !nce to discov] ! "t at ! m]idian f :i* ! longitude is measur$2 &1 by comp>+ t "t ) ! "t kn[n or discov]able 9 8 actual situation1 he inf]s 8 longitude4 ,b n only does ! pr$iction ( ! position ( ! moon1 ) respect to ^! cele/ial objects1 require a va/ numb] ( num]ical tables1 b likewise ! obs]vations necess>y to be made by ! m>9]1 9 ord] to det]m9e ! lun> di/ances1 also require sev]al tables4 ,to pr$ict ! exact position ( any fix$ />1 requires n less ?an t5 num]ical tables peculi> to t />2 & if ! m>9] be furni%$ 7z is actually ! case7 ) tables ( ! pr$ict$ di/ances ( ! moon f "o hundr$ su* />s1 su* pr$ictions mu/ require n less ?an a ?\s& num]ical tables4 ,reg>d+ ! range ( ! moon ?r\gh ! firmam5t1 h[ev]1 x w readily be conceiv$ t a hundr$ />s =m b a scanty supply2 especially :5 x is consid]$ t an accurate me?od ( det]m9+ ! longitude1 consi/s 9 obs]v+ ! ext9ction ( a /> by ! d>k edge ( ! moon4 ,)9 ! limits ( ! lun> orbit !re are n less ?an "o ?\s& />s1 :i* are s situat$ z to be 9 ! moon's pa?1 & !re=e to exhibit1 at "s p]iod or o!r1 ^? desirable occultations4 ,^! />s are also ( su* magnitudes1 t _! occultations may be di/9ctly obs]v$ f ! deck1 ev5 :5 subject to all ! un/ead9ess produc$ by an agitat$ sea4 ,to pr$ict ! occultations ( su* />s1 w\ld require n less ?an t5 ?\s& tables4 ,! />s f :i* lun> di/ances might be tak5 are /ill m num]\s2 & we may safely pron\nce1 t1 great z has be5 ! improvem5t effect$ rec5tly 9 our ,nautical ,almanac1 x does n yet furni% m ?an a small fraction ( t aid to navigation 79 ! l>ge s5se ( t t]m71 :i*1 ) great] facility1 exp$ition1 & economy 9 ! calculation & pr9t+ ( tables1 x might be made to supply4 ,tables necess>y to det]m9e ! places ( ! planets are n less necess>y ?an ^? = ! sun1 moon1 & />s4 ,"s notion ( ! numb] & complexity ( ^! tables may be =m$1 :5 we /ate t ! positions ( ! two pr9cipal planets1 7& ^! ! mo/ necess>y = ! navigator17 ,jupit] & ,saturn1 require ea* n less ?an "o hundr$ & sixte5 tables4 ,yet x is n only necess>y to pr$ict ! position ( ^! bodies1 b x is likewise exp$i5t to tabulate ! motions ( ! f\r satellites ( ,jupit]1 to pr$ict ! exact times at :i* !y ent] 8 %ad[1 & at :i* _! %ad[s cross 8 disc1 z well z ! times at :i* !y are int]pos$ betwe5 him & ! ,e>?1 & he betwe5 !m & ! ,e>?4 ,among ! ext5sive classes ( tables "h enum]at$1 !re are sev]al :i* are 9 _! nature p]man5t & unalt]able1 & w\ld nev] require to be recomput$1 if !y c\ld once be comput$ ) p]fect accuracy on accurate data2 b ! data on :i* su* computations are conduct$1 c only be reg>d$ z approximations to tru?1 )9 limits ! ext5t ( :i* mu/ necess>ily v>y ) our k ( a/ronomical sci5ce4 ,x has accord+ly happ5$1 t "o set ( tables aft] ano!r has be5 sup]s$$ ) ea* advance ( a/ronomical sci5ce4 ,"s /rik+ examples ( ?is may n be un9/ructive4 ,9 #agfe1 ! ,bo>d ( ,longitude paid to ! celebrat$ ,eul] ! sum ( ,l4#cjj1 = furni%+ g5]al =mul = ! computation ( lun> tables4 ,pr(essor ,may] 0 employ$ to calculate ! tables ^u ^! =mul1 & ! sum ( ,l4#cjjj 0 vot$ = !m by ! ,briti% ,p>liam5t1 to 8 wid[1 aft] 8 decease4 ,^! tables _h be5 us$ = t5 ye>s1 f #agff to #aggf1 9 comput+ ! ,nautical ,almanac1 :5 !y 7 sup]s$$ by new & improv$ tables1 compos$ by ,mr ,*>les ,mason1 "u ! direction ( ,dr ,maskelyne1 f calculations made by ord] ( ! ,bo>d ( ,longitude1 on ! obs]vations ( ,dr ,bradley4 ,a f>!r improvem5t 0 made by ,mason 9 #aghj2 b a mu* m ext5sive improvem5t took place 9 ! lun> calculations by ! publication ( ! tables ( ! ,moon1 by ,m4 ,brg1 d$uc$ f ,laplace's !ory1 9 #ahjf4 ,p]fect1 h[ev]1 z ,brg's tables 7 consid]$1 at ! "t ( _! publication1 !y 71 )9 ! %ort p]iod ( six ye>s1 sup]s$$ by a m accurate set ( tables publi%$ by ,burckh>dt 9 #ahab2 & ^! also h s9ce be5 foll[$ by ! tables ( ,damoiseau4 ,pr(essor ,s*uma*] has calculat$ by ! latt] tables 8 ephem]is ( ! ,planet>y ,lun> ,di/ances1 & a/ronom]s w h5ce be enabl$ to put to ! /rict te/ ( obs]vation ! m]its ( ! tables ( ,burckh>dt & ,damoiseau4#f ,footnote #f3 ,a comp>ison ( ! results = #ahcd1 w be f\nd 9 ! ,nautical ,almanac = #ahce4 ,! sol> tables h und]gone1 f "t to "t1 simil> *anges4 ,! sol> tables ( ,may] 7 us$ 9 ! computation ( ! ,nautical ,almanac1 f its comm5cem5t 9 #agfg1 to #ahjd inclusive4 ,)9 ! six ye>s imm$iately succe$+ #ahjd1 n less ?an ?ree successive sets ( sol> tables appe>$1 ea* improv+ on ! o!r2 ! fir/ by ,b>on de ,za*1 ! second by ,delambre1 "u ! direction ( ! ,fr5* ,bo>d ( ,longitude1 & ! ?ird by ,c>l9i4 ,! la/1 h[ev]1 diff] only 9 arrangem5t f ^? ( ,delambre4 ,simil> obs]vations w be applicable to ! tables ( ! pr9cipal planets4 ,b\v>d publi%$1 9 #ahjc1 tables ( ,jupit] & ,saturn2 b f ! improv$ /ate ( a/ronomy1 he f\nd x necess>y to recompute ^! tables 9 #ahba4 ,al?\gh x is n[ ab\t ?irty ye>s s9ce ! discov]y ( ! f\r new planets1 ,c]es1 ,pallas1 ,juno1 & ,ve/a1 x 0 n till rec5tly t tables ( _! motions 7 publi%$4 ,!y h lately appe>$ 9 ,encke's ,ephem]is4 ,we h ?us attempt$ to convey "s notion 7?\gh necess>ily a v inadequate "o7 ( ! imm5se ext5t ( num]ical tables :i* x has be5 f\nd necess>y to calculate & pr9t = ! purposes ( ! arts & sci5ces4 ,we h be=e u a catalogue ( ! tables conta9$ 9 ! libr>y ( "o private individual1 consi/+ ( n less ?an "o hundr$ & =ty volumes4 ,among ^! !re are no duplicate copies3 & we obs]ve t _m ( ! mo/ celebrat$ volum9\s tabul> works are n conta9$ among !m4 ,!y are conf9$ exclusively to ari?metical & trigonometrical tables2 &1 consequ5tly1 ! myriad ( a/ronomical & nautical tables are totally exclud$ f !m4 ,nev]!less1 !y conta9 an ext5t ( pr9t$ surface cov]$ ) figures am\nt+ to above sixte5 ?\s& squ>e feet4 ,we h tak5 at r&om =ty ( ^! tables1 & h f\nd t ! numb] ( errors ackn[l$g$ 9 ! respective errata1 am\nts to above ?ree ?\s& sev5 hundr$4 ,to be conv9c$ ( ! necessity :i* has exi/$ = accurate num]ical tables1 x w only be necess>y to consid] at :at an imm5se exp5diture ( lab\r & ( money ev5 ! imp]fect ones :i* we possess h be5 produc$4 ,to enable ! read] to e/imate ! difficulties :i* att5d ! atta9m5t ev5 ( a limit$ degree ( accuracy1 we %all n[ expla9 "s ( ! exp$i5ts :i* h be5 f "t to "t resort$ to = ! atta9m5t ( num]ical correctness 9 calculat+ & pr9t+ !m4 ,among ! sci5tific ent]prises :i* ! ambition ( ! ,fr5* nation aspir$ to dur+ ! ,republic1 0 ! con/ruction ( a magnific5t sy/em ( num]ical tables4 ,_! mo/ di/+ui%$ ma!maticians 7 call$ ^u to contribute to ! atta9m5t ( ?is important object2 & ! sup]9t5d5ce ( ! und]tak+ 0 confid$ to ! celebrat$ ,prony1 :o co-op]at$ ) ! gov]nm5t 9 ! adoption ( su* means z might be expect$ to ensure ! production ( a sy/em ( log>i?mic & trigonometric tables1 con/ruct$ ) su* accuracy t !y %\ld =m a monum5t ( calculation ! mo/ va/ & impos+ t _h ev] be5 execut$1 or ev5 conceiv$4 ,to accompli% ?is gigantic task1 ! pr9ciple ( ! division ( lab\r1 f\nd to be s p[]ful 9 manufactures1 0 resort$ to ) s+ul> success4 ,! p]sons employ$ 9 ! "w 7 divid$ into ?ree sections3 ! fir/ consi/$ ( half a doz5 ( ! mo/ em95t analy/s4 ,_! duty 0 to inve/igate ! mo/ conv5i5t ma!matical =mul1 :i* %\ld enable ! comput]s to proce$ ) ! greate/ exp$ition & accuracy by ! me?od ( ,diff]5ces1 ( :i* we %all speak m fully h]eaft]4 ,^! =mul1 :5 decid$ ^u by ?is fir/ section1 7 h&$ ov] to ! second section1 :i* consi/$ ( eight or t5 prop]ly qualifi$ ma!maticians4 ,x 0 ! duty ( ?is second section to conv]t into numb]s c]ta9 g5]al or algebraical expressions :i* occurr$ 9 ! =mul1 s z to prep>e !m = ! h&s ( ! comput]s4 ,?us prep>$1 ^! =mul 7 h&$ ov] to ! ?ird section1 :o =m$ a body ( ne>ly "o hundr$ comput]s4 ,! duty ( ?is num]\s section 0 to compute ! numb]s f9ally int5d$ = ! tables4 ,e possible precaution 0 ( c\rse tak5 to ensure ! num]ical accuracy ( ! results4 ,ea* numb] 0 calculat$ by two or m di/9ct & indep5d5t comput]s1 & its tru? & accuracy det]m9$ by ! co9cid5ce ( ! results ?us obta9$4 ,! body ( tables ?us calculat$ occupi$ 9 manuscript sev5te5 folio volumes4#g ,footnote #g3 ,^! tables 7 nev] publi%$4 ,! pr9t+ ( !m 0 comm5c$ by ,didot1 & a small portion 0 actually /]eotyp$1 b nev] publi%$4 ,soon aft] ! comm5cem5t ( ! und]tak+1 ! sudd5 fall ( ! assignats r5d]$ x impossible = ,didot to fulfil 8 contract ) ! gov]nm5t4 ,! "w 0 accord+ly ab&on$1 & has nev] s9ce be5 resum$4 ,we h be=e u a copy ( #ajj pages folio ( ! portion :i* 0 pr9t$ at ! "t ! "w 0 /opp$1 giv5 to a fri5d on a late occasion by ,didot himself4 ,x 0 rem>k$ 9 ?is1 z 9 o!r simil> cases1 t ! comput]s :o committ$ fewe/ errors 7 ^? :o und]/ood no?+ beyond ! process ( addition4 ,z an example ( ! precautions :i* h be5 consid]$ necess>y to gu>d aga9/ errors 9 ! calculation ( num]ical tables1 we %all fur!r /ate ^? :i* 7 adopt$ by ,mr ,babbage1 previ\sly to ! publication ( 8 tables ( log>i?ms4 ,9 ord] to r5d] ! t]m9al figure ( tables 9 :i* "o or m decimal places are omitt$ z accurate z x c be1 x has be5 ! practice to compute "o or m ( ! succe$+ figures2 & if ! fir/ omitt$ figure be great] ?an #d1 !n ! t]m9al figure is always increas$ by #a1 s9ce ! value ( ! tabulat$ numb] is by su* means br\ght ne>] to ! tru?4#h ,! tables ( ,callet1 :i* 7 among ! mo/ accurate publi%$ log>i?ms1 & :i* ext5d$ to sev5 places ( decimals1 7 fir/ c>efully comp>$ ) ! tables ( ,vega1 :i* ext5d$ to t5 places1 9 ord] to discov] :e!r ,callet _h made ! above correction ( ! f9al figure 9 e case :]e x 0 necess>y4 ,?is previ\s precaution be+ tak51 & ! corrections :i* appe>$ to be necess>y be+ made 9 a copy ( ,callet's tables1 ! pro(s ( ,mr ,babbage's tables 7 submitt$ to ! foll[+ te/3 ,!y 7 fir/ comp>$1 numb] by numb]1 ) ! correct$ copy ( ,callet's log>i?ms2 secondly1 ) ,hutton's log>i?ms2 & ?irdly1 ) ,vega's log>i?ms4 ,! corrections ?us sugge/$ be+ m>k$ 9 ! pro(s1 correct$ revises 7 receiv$ back4 ,^! revises 7 !n aga9 comp>$1 numb] by numb]1 fir/ ) ,vega's log>i?ms2 secondly1 ) ! log>i?ms ( ,callet2 & ?irdly1 z f> z ! fir/ #bj1jjj numb]s1 ) ! correspond+ ones 9 ,briggs's log>i?ms4 ,!y 7 n[ return$ to ! pr9t]1 & 7 /]eotyp$2 pro(s 7 tak5 f ! /]eotyp$ plates1 :i* 7 put ?r\gh ! foll[+ ordeal3 ,!y 7 fir/ comp>$ once m ) ! log>i?ms ( ,vega z f> z #dg1ejj2 !y 7 !n comp>$ ) ! :ole ( ! log>i?ms ( ,g>dn]2 & next ) ! :ole ( ,taylor's log>i?ms2 & z a la/ te/1 !y 7 transf]r$ to ! h&s ( a diff]5t set ( read]s1 & 7 once m comp>$ ) ,taylor4 ,t ^! precautions 7 by no means sup]flu\s may be collect$ f ! foll[+ circum/ances m5tion$ by ,mr ,babbage3 ,9 ! %eets read imm$iately previ\s to /]eotyp+1 ?irty-two errors 7 detect$2 aft] /]eotyp+1 eight m 7 f\nd1 & correct$ 9 ! plates4 ,footnote #h3 ,?us suppose ! numb] express$ at full l5g? 7 #c4adaeibg4 ,if ! table ext5d to no m ?an f\r places ( decimals1 we %\ld tabulate ! numb] #c4adaf & n #c4adae4 ,! =m] w\ld be evid5tly ne>] to ! true numb] #c4adaeibg4 ,by su* elaborate & exp5sive precautions _m ( ! errors ( computation & pr9t+ may c]ta9ly be remov$2 b x is too mu* to expect t 9 g5]al su* measures c be adopt$2 & we accord+ly f9d by f> ! great] numb] ( tables disfigur$ by errors1 ! ext5t ( :i* is r to be conjectur$ ?an det]m9$4 ,:5 ! nature ( a num]ical table is consid]$1--page aft] page d5sely cov]$ ) figures1 & ) no?+ else1--! *ances aga9/ ! detection ( any s+le error w be easily compreh5d$2 & x may !re=e be fairly presum$1 t = "o error :i* may happ5 to be detect$1 !re mu/ be a great numb] :i* escape detection4 ,not)/&+ ?is difficulty1 x is truly surpris+ h[ great a numb] ( num]ical errors h be5 detect$ by individuals no o!rwise conc]n$ 9 ! tables ?an 9 _! use4 ,mr ,baily /ates t he has himself detect$ 9 ! sol> & lun> tables1 f :i* our ,nautical ,almanac 0 = a long p]iod comput$1 m ?an five hundr$ errors4 ,9 ! multiplication table already m5tion$1 comput$ by ,dr ,hutton = ! ,bo>d ( ,longitude1 a s+le page 0 exam9$ & recomput$3 x 0 f\nd to conta9 ab\t =ty errors4 ,9 ord] to make ! calculations ^u ! numb]s f\nd 9 ! ,ephem]al ,tables publi%$ 9 ! ,nautical ,almanac1 x is necess>y t ! m>9] %\ld be suppli$ ) c]ta9 p]man5t tables4 ,a volume ( ^!1 to ! numb] ( ab\t ?irty1 0 accord+ly comput$1 & publi%$ at national exp5se1 by ord] ( ! ,bo>d ( ,longitude1 entitl$ ',tables requisite to be us$ ) ! ,nautical ,ephem]is = f9d+ ! latitude & longitude at sea4' ,9 ! fir/ edition ( ^! requisite tables1 !re 7 detect$1 by "o individual1 above a ?\s& errors4 ,! tables publi%$ by ! ,bo>d ( ,longitude = ! correction ( ! obs]v$ di/ances ( ! moon f c]ta9 fix$ />s1 are foll[$ by a table ( ackn[l$g$ errata1 ext5d+ to sev5 folio pages1 & conta9+ m ?an elev5 hundr$ errors4 ,ev5 ?is table ( errata itself is n correct3 a consid]able numb] ( errors h be5 detect$ 9 x1 s t errata ^u errata h become necess>y4 ,"o ( ! te/s mo/ frequ5tly resort$ to = ! detection ( errors 9 num]ical tables1 has be5 ! comp>ison ( tables ( ! same k9d1 publi%$ by diff]5t au?ors4 ,x has be5 g5]ally consid]$ t ^? numb]s 9 :i* !y are f\nd to agree mu/ be correct2 inasmu* z ! *ances are suppos$ to be v consid]able aga9/ two or m indep5d5t comput]s fall+ into precisely ! same errors4 ,h[ f> ?is co9cid5ce may be safely assum$ z a te/ ( accuracy we %all pres5tly see4 ,a few ye>s ago1 x 0 f\nd desirable to compute "s v accurate log>i?mic tables = ! use ( ! great national survey ( ,irel&1 :i* 0 !n1 & /ill is 9 progress2 & on t occasion a c>eful comp>ison ( v>i\s log>i?mic tables 0 made4 ,six rem>kable errors 7 detect$1 :i* 7 f\nd to be common to sev]al app>5tly indep5d5t sets ( tables4 ,?is s+ul> co9cid5ce l$ to an unusually ext5sive exam9ation ( ! log>i?mic tables publi%$ bo? 9 ,engl& & 9 o!r c\ntries2 by :i* x appe>$ t ?irte5 sets ( tables1 publi%$ 9 ,london betwe5 ! ye>s #afcc & #ahbb1 all agre$ 9 ^! six errors4 ,^u ext5d+ ! enquiry to =eign tables1 x appe>$ t two sets ( tables publi%$ at ,p>is1 "o at ,g\da1 "o at ,avignon1 "o at ,b]l91 & "o at ,flor5ce1 7 infect$ by exactly ! same six errors4 ,! only tables :i* 7 f\nd free f !m 7 ^? ( ,vega1 & ! m rec5t impressions ( ,callet4 ,x happ5$ t ! ,royal ,society possess$ a set ( tables ( log>i?ms pr9t$ 9 ! ,*9ese "*1 & on ,*9ese pap]1 consi/+ ( two volumes3 ^! volumes conta9$ no indication or ackn[l$gm5t ( be+ copi$ f any o!r "w4 ,!y 7 exam9$2 & ! result 0 ! detection 9 !m ( ! same six errors4#i ,footnote #i3 ,memoirs ,a/4 ,soc4 vol4 ,,iii1 p4 #fe4 ,x is q app>5t t ?is rem>kable co9cid5ce ( error mu/ h aris5 f ! v>i\s tables be+ copi$ successively "o f ano!r4 ,! e>lie/ "w 9 :i* !y appe>$ 0 ,vlacq's ,log>i?ms1 7folio1 ,g\da1 #afbh72 & f x1 d\btless1 ^? :i* imm$iately succe$$ x 9 po9t ( "t 7 copi$2 f :i* ! same errors 7 subsequ5tly transcrib$ into all ! o!r1 includ+ ! ,*9ese log>i?ms4 ,! mo/ c]ta9 & effectual *eck ^u errors :i* arise 9 ! process ( computation1 is to cause ! same computations to be made by sep>ate & indep5d5t comput]s2 & ?is *eck is r5d]$ /ill m decisive if !y make _! computations by diff]5t me?ods4 ,x is1 nev]!less1 a rem>kable fact1 t sev]al comput]s1 work+ sep>ately & indep5d5tly1 d frequ5tly commit precisely ! same error2 s t falsehood 9 ?is case assumes t "* ( consi/5cy1 :i* is reg>d$ z ! exclusive attribute ( tru?4 ,in/ances ( ?is are famili> to mo/ p]sons :o h _h ! managem5t ( ! computation ( tables4 ,we h reason to "k1 t ,m4 ,prony exp]i5c$ x on _m occasions 9 ! managem5t ( ! great ,fr5* tables1 :5 he f\nd ?ree1 & ev5 a great] numb] ( comput]s1 work+ sep>ately & indep5d5tly1 to return him ! same num]ical result1 & t result wrong4 ,mr ,/rat=d1 ! conductor ( ! ,nautical ,almanac1 to ^: tal5ts & zeal t "w owes ! execution ( its rec5t improvem5ts1 has m ?an once obs]v$ a simil> occurr5ce4 ,b "o ( ! mo/ signal examples ( ?is k9d1 ( :i* we are aw>e1 is relat$ by ,mr ,baily4 ,! catalogue ( />s publi%$ by ! ,a/ronomical ,society 0 comput$ by two sep>ate & indep5d5t p]sons1 & 0 aft]w>ds comp>$ & exam9$ ) great c>e & att5tion by ,mr ,/rat=d4 ,on exam9+ ?is catalogue1 & recalculat+ a portion ( x1 ,mr ,baily discov]$ an error 9 ! case ( ! />1 ,cephei4 ,its "r asc5sion 0 calculat$ wrongly1 & yet consi/5tly1 by two comput]s work+ sep>ately4 ,_! num]ical results agre$ precisely 9 e figure2 & ,mr ,/rat=d1 on exam9+ ! catalogue1 fail$ to detect ! error4 ,mr ,baily hav+ reason1 f "s discordancy :i* he obs]v$1 to suspect an error1 recomput$ ! place ( ! /> ) a view to discov] x2 & he himself1 9 ! fir/ in/ance1 obta9$ precisely ! same errone\s num]ical result4 ,x 0 only on go+ ov] ! op]ation a second "t t he accid5tally discov]$ t he _h inadv]t5tly committ$ ! same error4#aj ,footnote #aj3 ,memoirs ,a/4 ,soc4 vol4 iv41 p4 #bij4 ,x appe>s1 !re=e1 t ! co9cid5ce ( diff]5t tables1 ev5 :5 x is c]ta9 t !y c\ld n h be5 copi$ "o f ano!r1 b mu/ h be5 comput$ indep5d5tly1 is n a decisive te/ ( _! correctness1 nei!r is x possible to ensure accuracy by ! device ( sep>ate & indep5d5t computation4 ,besides ! errors incid5tal to ! process ( computation1 !re are fur!r liabilities 9 ! process ( transcrib+ ! f9al results ( ea* calculation into ! fair copy ( ! table design$ = ! pr9t]4 ,! next s\rce ( error lies ) ! compositor1 9 transf]r+ ?is copy into type4 ,b ! liabilities to error d n /op ev5 "h2 = x frequ5tly happ5s1 t aft] ! press has be5 fully correct$1 errors w be produc$ 9 ! process ( pr9t+4 ,a rem>kable in/ance ( ?is occurs 9 "o ( ! six errors detect$ 9 s _m diff]5t tables already m5tion$4 ,9 "o ( ^! cases1 ! la/ five figures ( two successive numb]s ( a log>i?mic table 7 ! foll[+3-- #cehge #ajdcf4 ,n[1 bo? ( ^! are errone\s2 ! figure #h 9 ! fir/ l9e %\ld be #d1 & ! figure #d 9 ! second %\ld be #h4 ,x is evid5t t ! types1 z fir/ compos$1 7 correct2 b 9 ! c\rse ( pr9t+1 ! two types #d & #h be+ loose1 adh]$ to ! ink+-balls1 & 7 drawn out3 ! pressm5 9 replac+ !m transpos$ !m1 putt+ ! #h above & ! #d bel[1 in/ead ( vice v]sa4 ,x w\ld be a curi\s enquiry1 7 x possible to obta9 all ! copies ( ! orig9al edition ( ,vlacq's ,log>i?ms1 publi%$ at ,g\da 9 #afbh1 f :i* ?is error appe>s to h be5 copi$ 9 all ! subsequ5t tables1 to asc]ta9 :e!r x ext5ds ?r\gh ! entire edition4 ,x w\ld probably1 nay almo/ c]ta9ly1 be discov]$ t "s ( ! copies ( t edition are correct 9 ?is numb]1 :ile o!rs are incorrect2 ! =m] hav+ be5 work$ (f be=e ! transposition ( ! types4 ,x is a circum/ance wor?y ( notice1 t ?is error 9 ,vlacq's tables has produc$ a correspond+ error 9 a v>iety ( o!r tables d$uc$ f !m1 9 :i* nev]!less ! errone\s figures 9 ,vlacq are omitt$4 ,9 no less ?an sixte5 sets ( tables publi%$ at v>i\s times s9ce ! publication ( ,vlacq1 9 :i* ! log>i?ms ext5d only to sev5 places ( figures1 ! error j m5tion$ 9 ! eigh? place 9 ,vlacq causes a correspond+ error 9 ! sev5? place4 ,:5 ! la/ ?ree figures are omitt$ 9 ! fir/ ( ! above numb]s1 ! sev5? figure %\ld be #e1 inasmu* z ! fir/ ( ! omitt$ figures is "u #e3 ! errone\s ins]tion1 h[ev]1 ( ! figure #h 9 ,vlacq has caus$ ! figure #f to be sub/itut$ = #e 9 ! v>i\s tables j allud$ to4 ,= ! same reason1 ! errone\s occurr5ce ( #d 9 ! second numb] has caus$ ! adoption ( a #j in/ead ( a #a 9 ! sev5? place 9 ! o!r tables4 ,! only tables 9 :i* ?is error does n occur are ^? ( ,vega1 ! m rec5t editions ( ,callet1 & ! /ill lat] ,log>i?ms ( ,mr ,babbage4 ,! ,opus ,palat9um1 a "w publi%$ 9 #aeif1 conta9+ an ext5sive collection ( trigonometrical tables1 af=ds a rem>kable in/ance ( a tabul> error2 :i*1 z x is n g5]ally kn[n1 x may n be un9t]e/+ to m5tion "h4 ,aft] t "w _h be5 = sev]al ye>s 9 circulation 9 e "p ( ,europe1 x 0 discov]$ t ! comm5cem5t ( ! table ( co-tang5ts & co-secants 0 vitiat$ by an error ( consid]able magnitude4 ,9 ! fir/ co-tang5t ! la/ n9e places ( figures 7 incorrect2 b f ! mann] 9 :i* ! numb]s ( ! table 7 comput$1 ! error 0 gradually1 ?\gh sl[ly1 dim9i%$1 until at l5g? x became ext+ui%$ 9 ! eighty-six? page4 ,aft] ! detection ( ?is ext5sive error1 ,pitiscus und]took ! recomputation ( ! eighty-six errone\s pages4 ,8 correct$ calculation 0 pr9t$1 & ! errone\s "p ( ! rema9+ copies ( ! ,opus ,palat9um 0 cancell$4 ,b z ! correct$ table ( ,pitiscus 0 n publi%$ until #afjg1--?irte5 ye>s aft] ! orig9al "w1--! errone\s "p ( ! volume 0 cancell$ 9 comp>atively few copies1 & consequ5tly correct copies ( ! "w are n[ exce$+ly r>e4 ,?us1 9 ! collection ( tables publi%$ by ,m4 ,s*ulze1#aa ! :ole ( ! errone\s "p ( ! ,opus ,palat9um has be5 adopt$2 he hav+ us$ ! copy ( t "w :i* exi/s 9 ! libr>y ( ! ,academy ( ,b]l91 & :i* is "o ( ^? copies 9 :i* ! incorrect "p 0 n cancell$4 ,! correct$ copies ( ?is "w may be v easily di/+ui%$ at pres5t f ! errone\s ones3 x happ5$ t ! =m] 7 pr9t$ ) a v bad & worn-out type1 & ^u pap] ( a quality inf]ior to t ( ! orig9al "w4 ,on comp>+ ! fir/ eighty-six pages ( ! volume ) ! succe$+ ones1 !y are1 !re=e1 imm$iately di/+ui%able 9 ! correct$ copies4 ,besides ?is te/1 !re is ano!r1 :i* x may n be un9t]e/+ to po9t out3--,at ! bottom ( page #g 9 ! correct$ copies1 !re is an error 9 ! position ( ! words basis & hypo!nusa1 _! places be+ int]*ang$4 ,9 ! orig9al uncorrect$ "w ?is error does n exi/4 ,footnote #aa3 ,recueil des ,tables ,log>i?miques et ,trigonometriques4 ,p> ,j4 ,c4 ,s*ulze4 #b vols4 ,b]l93 #aggh4 ,at ! "t :5 ! calculation & publication ( ,taylor's ,log>i?ms 7 und]tak51 x s happ5$ t a simil> "w 0 9 progress 9 ,france2 & x 0 n until ! calculation ( ! ,fr5* "w 0 complet$1 t its au?or 0 in=m$ ( ! publication ( ! ,engli% "w4 ,?is circum/ance caus$ ! ,fr5* calculator to rel9qui% ! publication ( 8 tables4 ,! manuscript subsequ5tly pass$ into ! libr>y ( ,delambre1 &1 aft] 8 dea?1 0 pur*as$ at ! sale ( 8 books1 by ,mr ,babbage1 9 ^: possession x n[ is4 ,"s ye>s ago x 0 ?\ght advisable to comp>e ^! manuscript tables ) ,taylor's ,log>i?ms1 ) a view to asc]ta9 ! errors 9 ea*1 b especially 9 ,taylor4 ,! two works 7 peculi>ly well suit$ = ! atta9m5t ( ?is end2 z ! circum/ances "u :i* !y 7 produc$1 r5d]$ x q c]ta9 t !y 7 comput$ indep5d5tly ( ea* o!r4 ,! comp>ison 0 conduct$ "u ! direction ( ! late ,dr ,"y1 & ! result 0 ! detection ( ! foll[+ n9ete5 errors 9 ,taylor's ,log>i?ms4 ,to enable ^? :o us$ ,taylor's ,log>i?ms to make ! necess>y corrections 9 !m1 ! corrections ( ! detect$ errors appe>$ z foll[s 9 ! ,nautical ,almanac = #ahcb4 ,,errata1 detect$ 9 ,taylor's ,log>i?ms4 ,london3 #dto1 #agib4 ' #a ,e ,co-tang5t ( #a4ce4ce = #dcfga read #dbfga #b ,m ,co-tang5t ( #d4 #d4di --- #ffigf ---- #ffigi #c ,s9e ( #d4bc4ch --- #dcajg ---- #dcjjg #d ,s9e ( #d4bc4ci --- #dccha ---- #dcbha #e ,s ,s9e ( #f4de4eb --- #ajjja ---- #aajja #f ,kk ,co-s9e ( #ad4ah4 #c --- #ccih ---- #cbih #g ,ss ,tang5t ( #ah4 #a4ef --- #ejfd ---- #fjfd #h ,aaa ,co-tang5t ( #ba4aa4ad --- #fjfb ---- #eifb #i ,ggg ,tang5t ( #bc4dh4ai --- #fjhg ---- #eihg #aj ,co-tang5t ( #bc4dh4ai --- #ciac ---- #djac #aa ,iii ,s9e ( #be4 #e4 #d --- #cagc ---- #cahc #ab ,s9e ( #be4 #e4 #e --- #cbah ---- #cbbh #ac ,s9e ( #be4 #e4 #f --- #cbfc ---- #cbgc #ad ,s9e ( #be4 #e4 #g --- #ccjh ---- #ccah #ae ,s9e ( #be4 #e4 #h --- #ccec ---- #ccfc #af ,s9e ( #be4 #e4 #i --- #ccih ---- #cdjh #ag ,qqq ,tang5t ( #bh4ai4ci --- #fcjb ---- #fdjb #ah #d;,h ,tang5t ( #ce4ee4ea --- #afha ---- #aeha #ai #d,k ,co-s9e ( #cg4bi4 #b --- #eejc ---- #efjc ,an error be+ detect$ 9 ?is li/ ( ,,errata1 we f9d1 9 ! ,nautical ,almanac = ! ye> #ahcc1 ! foll[+ ,,erratum ( ! ,,errata ( ,taylor's ,log>i?ms3-- ',9 ! li/ ( ,,errata detect$ 9 ,taylor's ,log>i?ms1 = cos4 #d #ah' #c1 read cos4 #ad #ah' #b4' ,"h1 h[ev]1 confusion is worse conf\nd$2 = a new error1 n be=e exi/+1 & ( mu* great] magnitude1 is introduc$6 ,x w be necess>y1 9 ! ,nautical ,almanac = #ahcf1 7t = #ahce is already publi%$17 to introduce ! foll[+3 ,,erratum ( ! ,,erratum ( ! ,,errata ( ,,taylor's ,log>i?ms4 ,= cos4 #d #ah' #c1 read cos4 #ad #ah' #c4 ,if pro( 7 want$ to e/abli% incontrov]tibly ! utt] impracticability ( preclud+ num]ical errors 9 works ( ?is nature1 we %\ld f9d x 9 ?is succession ( error ^u error1 produc$1 9 spite ( ! univ]sally ackn[l$g$ accuracy & assiduity ( ! p]sons at pres5t employ$ 9 ! con/ruction & managem5t ( ! ,nautical ,almanac4 ,x is only by ! me*anical fabrication ( tables t su* errors c be r5d]$ impossible4 ,on exam9+ ?is li/ ) att5tion1 we h be5 p>ticul>ly /ruck ) ! circum/ances 9 :i* ^! errors appe> to h orig9at$4 ,x is a rem>kable fact1 t ( ! above n9ete5 errors1 eighte5 h aris5 f mi/akes 9 c>ry+4 ,errors #e1 #g1 #aj1 #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ai1 h aris5 f a c>riage be+ neglect$2 & errors #a1 #c1 #d1 #f1 #h1 #i1 & #ah1 f a c>riage be+ made :]e none %\ld take place4 ,9 f\r cases1 namely1 errors #h1 #i1 #aj1 & #af1 ?is has caus$ two figures to be wrong4 ,! only error ( ! n9ete5 :i* appe>s to h be5 a press error is ! second2 :i* has evid5tly aris5 f ! type #i be+ accid5tally inv]t$1 & ?us becom+ a #f4 ,?is may h orig9at$ ) ! compositor1 b m probably x took place 9 ! press-"w2 ! type #i be+ accid5tally drawn out ( ! =m by ! ink+-ball1 z m5tion$ 9 a =m] case1 & on be+ re/or$ to its place1 inv]t$ by ! pressman4 ,!re are two cases among ! above errata1 9 :i* an error1 committ$ 9 ! calculation ( "o numb]1 has evid5tly be5 ! cause ( o!r errors4 ,9 ! ?ird erratum1 a wrong c>riage 0 made1 9 comput+ ! s9e ( #d #bc' #ch4 ,! next numb] ( ! table 0 vitiat$ by ?is error2 = we f9d ! next erratum to be 9 ! s9e ( #d #bc' #ci1 9 :i* ! figure simil>ly plac$ is #a 9 excess4 ,a /ill m ext5sive effect ( ?is k9d appe>s 9 errata #aa1 #ab1 #ac1 #ad1 #ae1 #af4 ,a c>riage 0 neglect$ 9 comput+ ! s9e ( #be #e' #d1 & ?is produc$ a correspond+ error 9 ! five foll[+ numb]s ( ! table1 :i* are ^? correct$ 9 ! five foll[+ errata4 ,?is frequ5cy ( errors aris+ 9 ! process ( c>ry+1 w\ld af=d a curi\s subject ( metaphysical speculation respect+ ! op]ation ( ! faculty ( memory4 ,9 ! ari?metical process1 ! memory is employ$ 9 a tw(old way2--9 asc]ta9+ ea* successive figure ( ! calculat$ result by ! recollection ( a table committ$ to memory at an e>ly p]iod ( life2 & by ano!r act ( memory1 9 :i* ! numb] c>ri$ f column to column is reta9$4 ,x is a curi\s fact1 t ?is latt] circum/ance1 occurr+ only ! mom5t be=e1 & be+ 9 its nature little complex1 is s mu* m liable to be =gott5 or mi/ak5 ?an ! results ( r complicat$ tables4 ,x appe>s1 t among ! above errata1 ! errors #e1 #g1 #aj1 #aa1 #ag1 #ai1 h be5 produc$ by ! comput] =gett+ a c>riage2 :ile ! errors #a1 #c1 #f1 #h1 #i1 #ah1 h be5 produc$ by 8 mak+ a c>riage improp]ly4 ,?us1 s f> z ! above li/ ( errata af=ds gr\nds = judg+1 x w\ld seem1 7contr>y to :at might be expect$17 t ! error by :i* improp] c>riages are made is z frequ5t z t by :i* necess>y c>riages are ov]look$4 ,we tru/ t we h succe$$ 9 prov+1 fir/1 ! great national & univ]sal utility ( num]ical tables1 by %[+ ! va/ numb] ( !m1 :i* h be5 calculat$ & publi%$2 secondly1 t m effectual means are necess>y to obta9 su* tables suitable to ! pres5t /ate ( ! arts1 sci5ces & comm]ce1 by %[+ t ! exi/+ supply ( tables1 va/ z x c]ta9ly is1 is /ill scanty1 & utt]ly inadequate to ! dem&s ( ! community2--t x is r5d]$ ineffici5t1 n only 9 quantity1 b 9 quality1 by its want ( num]ical correctness2 & t su* num]ical correctness is altoge!r unatta9able until "s m p]fect me?od be discov]$1 n only ( calculat+ ! num]ical results1 b ( tabulat+ ^!1--( r$uc+ su* tallies to type1 & ( pr9t+ t type s z to int]cept ! possibility ( error dur+ ! press-"w4 ,su* are ! ends :i* are propos$ to be atta9$ by ! calculat+ ma*9]y inv5t$ by ,mr ,babbage4 ,! b5efits to be d]iv$ f ?is inv5tion _c be m /rongly express$ ?an !y h be5 by ,mr ,colebrooke1 ,presid5t ( ! ,a/ronomical ,society1 on ! occasion ( pres5t+ ! gold m$al vot$ by t body to ,mr ,babbage3--',9 no dep>tm5t ( sci5ce1 or ( ! arts1 does ?is discov]y promise to be s em95tly useful z 9 t ( a/ronomy1 & its k9dr$ sci5ces1 ) ! v>i\s arts dep5d5t on !m4 ,9 none are computations m op]ose ?an ^? :i* a/ronomy 9 p>ticul> requires2--9 none are prep>atory facilities m ne$ful2--9 none is error m detrim5tal4 ,! practical a/ronom] is int]rupt$ 9 8 pursuit1 & div]t$ f 8 task ( obs]vation by ! irksome lab\rs ( computation1 or 8 dilig5ce 9 obs]v+ becomes ineffectual = want ( yet great] indu/ry ( calculation4 ,let ! aid :i* tables previ\sly comput$ af=d1 be furni%$ to ! utmo/ ext5t :i* me*anism has made atta9able ?r\gh ,mr ,babbage's inv5tion1 & ! mo/ irksome portion ( ! a/ronom]'s task is alleviat$1 & a fre% impulse is giv5 to a/ronomical rese>*4' ,! fir/ /ep 9 ! progress ( ?is s+ul> inv5tion 0 ! discov]y ( "s common pr9ciple :i* p]vad$ num]ical tables ( e description2 s t by ! adoption ( su* a pr9ciple z ! basis ( ! ma*9]y1 a correspond+ degree ( g5]ality w\ld be conf]r$ ^u its calculations4 ,among ! prop]ties ( num]ical functions1 sev]al ( a g5]al nature exi/2 & x 0 a matt] ( no ord9>y difficulty1 & requir+ no common skill1 to select "o :i* might1 9 all respects1 be pref]able to ! o!rs4 ,:e!r or n t :i* 0 select$ by ,mr ,babbage af=ds ! greate/ practical advantages1 w\ld be extremely difficult to decide--p]haps impossible1 unless "s o!r projector c\ld be f\nd possess$ ( suffici5t g5ius1 & su/a9$ by suffici5t en]gy ( m9d & "*1 to attempt ! inv5tion ( calculat+ ma*9]y on o!r pr9ciples4 ,! pr9ciple select$ by ,mr ,babbage z ! basis ( t "p ( ! ma*9]y :i* calculates1 is ! ,me?od ( ,diff]5ces2 & he has 9 fact lit]ally ?r[n ?is ma!matical pr9ciple into :eel-"w4 ,9 ord] to =m a notion ( ! nature ( ! ma*9]y1 x w be necess>y1 fir/ to convey to ! read] "s idea ( ! ma!matical pr9ciple j allud$ to4 ,a num]ical table1 ( :atev] k9d1 is a s]ies ( numb]s :i* possess "s common "*1 & :i* proce$ increas+ or decreas+ accord+ to "s g5]al law4 ,suppos+ su* a s]ies cont9ually to increase1 let u imag9e ea* numb] 9 x to be subtract$ f t :i* foll[s x1 & ! rema9d]s ?us successively obta9$ to be rang$ beside ! fir/1 s z to =m ano!r table3 ^! numb]s are call$ ! fir/ diff]5ces4 ,if we suppose ^! likewise to increase cont9ually1 we may obta9 a ?ird table f !m by a l process1 subtract+ ea* numb] f ! succe$+ "o3 ?is s]ies is call$ ! second diff]5ces4 ,by adopt+ a l me?od ( proce$+1 ano!r s]ies may be obta9$1 call$ ! ?ird diff]5ces2 & s on4 ,by cont9u+ ?is process1 we %all at l5g? obta9 a s]ies ( diff]5ces1 ( "s ord]1 m or less high1 accord+ to ! nature ( ! orig9al table1 9 :i* we %all f9d ! same numb] con/antly repeat$1 to :atev] ext5t ! orig9al table may h be5 cont9u$2 s t if ! next s]ies ( diff]5ces _h be5 obta9$ 9 ! same mann] z ! prec$+ ones1 e t]m ( x w\ld be #j4 ,9 "s cases ?is w\ld cont9ue to :atev] ext5t ! orig9al table might be c>ri$2 b 9 all cases a s]ies ( diff]5ces w\ld be obta9$1 :i* w\ld cont9ue con/ant = a v long succession ( t]ms4 ,z ! successive s]ieses ( diff]5ces are d]iv$ f ! orig9al table1 & f ea* o!r1 by subtraction1 ! same succession ( s]ies may be reproduc$ 9 ! o!r direction by addition4 ,b let u suppose t ! fir/ numb] ( ! orig9al table1 & ( ea* ( ! s]ies ( diff]5ces1 includ+ ! la/1 be giv53 all ! numb]s ( ea* ( ! s]ies may !nce be obta9$ by ! m]e process ( addition4 ,! second t]m ( ! orig9al table w be obta9$ by add+ to ! fir/ ! fir/ t]m ( ! fir/ diff]5ce s]ies2 9 l mann]1 ! second t]m ( ! fir/ diff]5ce s]ies w be obta9$ by add+ to ! fir/ t]m1 ! fir/ t]m ( ! ?ird diff]5ce s]ies1 & s on4 ,! second t]ms ( all ! s]ieses be+ ?us obta9$1 ! ?ird t]ms may be obta9$ by a l process ( addition2 & s ! s]ies may be cont9u$4 ,^! obs]vations w p]haps be r5d]$ m cle>ly intelligible :5 illu/rat$ by a num]ical example4 ,! foll[+ is ! comm5cem5t ( a s]ies ( ! f\r? p[]s ( ! natural numb]s3-- ,no4 ,table4 #a #a #b #af #c #ha #d #bef #e #fbe #f #abif #g #bdja #h #djif #i #fefa #aj #aj1jjj #aa #ad1fda #ab #bj1gcf #ac #bh1efa ,by subtract+ ea* numb] f ! succe$+ "o 9 ?is s]ies1 we obta9 ! foll[+ s]ies ( fir/ diff]5ces3 #ae #fe #age #cfi #fga #aaje #afie #bdfe #cdci #dfda #fjie #ghbe ,9 l mann]1 subtract+ ea* t]m ( ?is s]ies f ! succe$+ "o1 we obta9 ! foll[+ s]ies ( second diff]5ces3-- #ej #aaj #aid #cjb #dcd #eij #ggj #igd #abjb #aded #agcj ,proce$+ ) ?is s]ies 9 ! same way1 we obta9 ! foll[+ s]ies ( ?ird diff]5ces3-- #fj #hd #ajh #acb #aef #ahj #bjd #bbh #beb #bgf ,proce$+ 9 ! same way ) ^!1 we obta9 ! foll[+ = ! s]ies ( f\r? diff]5ces3-- #bd #bd #bd #bd #bd #bd #bd #bd #bd ,x appe>s1 !re=e1 t 9 ?is case ! s]ies ( f\r? diff]5ces consi/s ( a con/ant repetition ( ! numb] #bd4 ,n[1 a slight consid]ation ( ! succession ( ari?metical op]ations by :i* we h obta9$ ?is result1 w %[1 t by rev]s+ ! process1 we c\ld obta9 ! table ( f\r? p[]s by ! m]e process ( addition4 ,beg9n+ ) ! fir/ numb]s 9 ea* successive s]ies ( diff]5ces1 & designat+ ! table & ! successive diff]5ces by ! lett]s ,t1 ,d#a ,d#b ,d#c ,d#d1 we h !n ! foll[+ to beg9 )3-- ,t ,d#a ,d#b ,d#c ,d#d #a #ae #ej #fj #bd ,add+ ea* numb] to ! numb] on its left1 & repeat+ #bd1 we get ! foll[+ z ! second t]ms ( ! sev]al s]ies3-- ,t ,d#a ,d#b ,d#c ,d#d #af #fe #aaj #hd #bd ,&1 9 ! same mann]1 ! ?ird & succe$+ t]ms z foll[s3-- ,no4 ,t ,d#a ,d#b ,d#c ,d#d #a #a #ae #ej #fj #bd #b #af #fe #aaj #hd #bd #c #ha #age #aid #ajh #bd #d #bef #cfi #cjb #acb #bd #e #fbe #fga #dcd #aef #bd #f #abif #aaje #eij #ahj #bd #g #bdja #afie #ggj #bjd #bd #h #djif #bdfe #igd #bbh #bd #i #fefa #cdci #abjb #beb #bd #aj #ajjjj #dfda #aded #bgf #aa #adfda #fjie #agcj #ab #bjgcf #ghbe #ac #bhefa ,!re are num]\s tables 9 :i*1 z already /at$1 to :atev] ord] ( diff]5ces we may proce$1 we %\ld n obta9 a s]ies ( rigor\sly con/ant diff]5ces2 b we %\ld always obta9 a c]ta9 numb] ( diff]5ces :i* to a giv5 numb] ( decimal places w\ld rema9 con/ant = a long succession ( t]ms4 ,x is pla9 t su* a table might be calculat$ by addition 9 ! same mann] z ^? :i* h a diff]5ce rigor\sly & cont9u\sly con/ant2 & if at e po9t :]e ! la/ diff]5ce requires an increase1 t increase be giv5 to x1 ! same pr9ciple ( addition may aga9 be appli$ = a l succession ( t]ms1 & s on4 ,by ?is pr9ciple x appe>s1 t all tables 9 :i* ea* s]ies ( diff]5ces cont9ually increases1 may be produc$ by ! op]ation ( addition alone2 provid$ ! fir/ t]ms ( ! table1 & ( ea* s]ies ( diff]5ces1 be giv5 9 ! fir/ in/ance4 ,b x sometimes happ5s1 t :ile ! table cont9ually increases1 "o or m s]ieses ( diff]5ces may cont9ually dim9i%4 ,9 ?is case1 ! s]ies ( diff]5ces are f\nd by subtract+ ea* t]m ( ! s]ies1 n f t :i* foll[s1 b f t :i* prec$es x2 & consequ5tly1 9 ! re-production ( ! sev]al s]ieses1 :5 _! fir/ t]ms are giv51 x w be necess>y 9 "s cases to obta9 !m by addition1 & 9 o!rs by subtraction4 ,x is possible1 h[ev]1 /ill to p]=m all ! op]ations by addition alone3 ?is is effect$ 9 p]=m+ ! op]ation ( subtraction1 by sub/itut+ = ! subtrah5d its ari?metical complem5t1 & add+ t1 omitt+ ! unit ( ! highe/ ord] 9 ! result4 ,?is process1 & its pr9ciple1 w be readily compreh5d$ by an example4 ,let x be requir$ to subtract #ceg f #gfh4 ,! common process w\ld be z foll[s3-- ,f #gfh ,subtract #ceg ---- ,rema9d] #daa ,! ari?metical complem5t ( #ceg1 or ! numb] by :i* x falls %ort ( #ajjj1 is #fdc4 ,n[1 if ?is numb] be add$ to #gfh1 & ! fir/ figure on ! left be /ruck out ( ! sum1 ! process w be z foll[s3-- ,to #gfh ,add #fdc ---- ,sum #adaa ---- ,rema9d] s\ght #daa ,! pr9ciple on :i* ?is process is f\nd$ is easily expla9$4 ,9 ! latt] process we h fir/ add$ #fdc1 & !n subtract$ #ajjj4 ,on ! :ole1 !re=e1 we h subtract$ #ceg1 s9ce ! numb] actually subtract$ exce$s ! numb] previ\sly add$ by t am\nt4 ,s9ce1 !re=e1 subtraction may be effect$ 9 ?is mann] by addition1 x foll[s t ! calculation ( all s]ieses1 s f> z an ord] ( diff]5ces c be f\nd 9 !m :i* cont9ues con/ant1 may be conduct$ by ! process ( addition alone4 ,x also appe>s f :at has be5 /at$1 t ea* addition consi/s only ( two op]ations4 ,h[ev] num]\s ! figures may be ( :i* ! sev]al pairs ( numb]s to be ?us add$ may consi/1 x is obvi\s t ! op]ation ( add+ !m c only consi/ ( repetitions ( ! process ( add+ "o digit to ano!r2 & ( c>ry+ "o f ! column ( inf]ior units to ! column ( units next sup]ior :5 necess>y4 ,if we w\ld !re=e r$uce su* a process to ma*9]y1 x w\ld only be necess>y to discov] su* a comb9ation ( mov+ p>ts z are capable ( p]=m+ ^! two processes ( add+ & c>ry+ on two s+le figures2 =1 ?is be+ once accompli%$1 ! process ( add+ two numb]s1 consi/+ ( any numb] ( digits1 w be effect$ by repeat+ ! same me*anism z (t5 z !re are pairs ( digits to be add$4 ,su* 0 ! simple =m to :i* ,mr ,babbage r$uc$ ! problem ( discov]+ ! calculat+ ma*9]y2 & we %all n[ proce$ to convey "s notion ( ! mann] 9 :i* he solv$ x4 ,= ! sake ( illu/ration1 we %all suppose t ! table to be calculat$ %all consi/ ( numb]s n exce$+ six places ( figures2 & we %all also suppose t ! diff]5ce ( ! fif? ord] is ! con/ant diff]5ce4 ,imag9e1 !n1 six r[s ( :eels1 ea* :eel c>ry+ ^u x a dial-plate l t ( a common clock1 b consi/+ ( t5 in/ead ( twelve divisions2 ! sev]al divisions be+ m>k$ #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #j4 ,let ^! dials be suppos$ to revolve :5ev] ! :eels to :i* !y are atta*$ are put 9 motion1 & to turn 9 su* a direction t ! s]ies ( increas+ numb]s %all pass "u ! index :i* appe>s ov] ea* dial3--?us1 aft] #j passes ! index1 #a foll[s1 !n #b1 #c1 & s on1 z ! dial revolves4 ,9 ,fig4 #a are repres5t$ six horizontal r[s ( su* dials4 ,fig4 #a4 ,! me?od ( diff]5ces1 z already expla9$1 requires1 t 9 proce$+ ) ! calculation1 ?is app>atus %\ld p]=m cont9ually ! addition ( ! numb] express$ ^u ea* r[ ( dials1 to ! numb] express$ ^u ! r[ imm$iately above x4 ,n[1 we %all fir/ expla9 h[ ?is process ( addition may be conceiv$ to be p]=m$ by ! motion ( ! dials2 & 9 do+ s1 we %all consid] sep>ately ! processes ( addition & c>riage1 consid]+ ! addition fir/1 & !n ! c>riage4 ,let u fir/ suppose ! l9e ,d#a to be add$ to ! l9e ,t4 ,to accompli% ?is1 let u imag9e t :ile ! dials on ! l9e ,d#a are quiesc5t1 ! dials on ! l9e ,t are put 9 motion1 9 su* a mann]1 t z _m divisions on ea* dial %all pass "u its index1 z !re are units 9 ! numb] at ! index imm$iately bel[ x4 ,x is evid5t t ?is condition supposes1 t if #j be at any index on ! l9e ,d#a1 ! dial imm$iately above x 9 ! l9e ,t %all n move4 ,n[ ! motion "h suppos$1 w\ld br+ "u ! indices on ! l9e ,t su* a numb] z w\ld be produc$ by add+ ! numb] ,d#a to ,t1 neglect+ all ! c>riages2 = a c>riage %\ld h tak5 place 9 e case 9 :i* ! figure #i ( any dial 9 ! l9e ,t _h pass$ "u ! index dur+ ! add+ motion4 ,to accompli% ?is c>riage1 x w\ld be necess>y t ! dial imm$iately on ! left ( any dial 9 :i* #i passes "u ! index1 %\ld be advanc$ "o division1 indep5d5tly ( ^? divisions :i* x may h be5 advanc$ by ! addition ( ! numb] imm$iately bel[ x4 ,?is effect may be conceiv$ to take place 9 ei!r ( two ways4 ,x may be ei!r produc$ at ! mom5t :5 ! division betwe5 #i & #j ( any dial passes "u ! index2 9 :i* case ! process ( c>ry+ w\ld g on simultane\sly ) ! process ( add+2 or ! process ( c>ry+ may be po/pon$ 9 e in/ance until ! process ( addition1 )\t c>ry+1 has be5 complet$2 & !n by ano!r di/9ct & indep5d5t motion ( ! ma*9]y1 a c>riage may be made by advanc+ "o division all ^? dials on ! "r ( :i* a dial _h1 dur+ ! previ\s addition1 pass$ f #i to #j "u ! index4 ,! latt] is ! me?od adopt$ 9 ! calculat+ ma*9]y1 9 ord] to enable its inv5tor to con/ruct ! c>ry+ ma*9]y indep5d5t ( ! add+ me*anism4 ,hav+ expla9$ ! motion ( ! dials by :i* ! addition1 exclud+ ! c>riages ( ! numb] on ! r[ ,d#a1 may be made to ! numb] on ! r[ ,t1 ! same explanation may be appli$ to ! numb] on ! r[ ,d#b to ! numb] on ! r[ ,d#a2 also1 ( ! numb] ,d#c to ! numb] on ! r[ ,d#b1 & s on4 ,n[ x is possible to suppose ! additions ( all ! r[s1 except ! fir/1 to be made to all ! r[s except ! la/1 simultane\sly2 & aft] ^! additions h be5 made1 to conceive all ! requisite c>riages to be also made by advanc+ ! prop] dials "o division =w>d4 ,?is w\ld suppose all ! dials 9 ! s*eme to receive _! add+ motion toge!r2 &1 ?is be+ accompli%$1 ! requisite dials to receive _! c>ry+ motions toge!r4 ,! production ( s great a numb] ( simultane\s motions ?r\gh\t any ma*9]y1 w\ld be att5d$ ) great me*anical difficulties1 if inde$ x be practicable4 ,9 ! calculat+ ma*9]y x is n attempt$4 ,! additions are p]=m$ 9 two successive p]iods ( "t1 & ! c>riages 9 two o!r p]iods ( "t1 9 ! foll[+ mann]4 ,we %all suppose "o complete revolution ( ! axis :i* moves ! ma*9]y1 to make "o complete set ( additions & c>riages2 x w !n make !m 9 ! foll[+ ord]3-- ,! fir/ qu>t] ( a turn ( ! axis w add ! second1 f\r?1 & six? r[s to ! fir/1 ?ird1 & fif?1 omitt+ ! c>riages2 ?is x w d by caus+ ! dials on ! fir/1 ?ird1 & fif? r[s1 to turn ?r\gh z _m divisions z are express$ by ! numb]s at ! indices bel[ !m1 z already expla9$4 ,! second qu>t] ( a turn w cause ! c>riages consequ5t on ! previ\s addition1 to be made by mov+ =w>d ! prop] dials "o division4 7,dur+ ^! two qu>t]s ( a turn1 ! dials ( ! fir/1 ?ird1 & fif? r[ alone h be5 mov$2 ^? ( ! second1 f\r?1 & six?1 h be5 quiesc5t47 ,! ?ird qu>t] ( a turn w produce ! addition ( ! ?ird & fif? r[s to ! second & f\r?1 omitt+ ! c>riages2 :i* x w d by caus+ ! dials ( ! second & f\r? r[s to turn ?r\gh z _m divisions z are express$ by ! numb]s at ! indices imm$iately bel[ !m4 ,! f\r? & la/ qu>t] ( a turn w cause ! c>riages consequ5t on ! previ\s addition1 to be made by mov+ ! prop] dials =w>d "o division4 ,?is evid5tly completes "o calculation1 s9ce all ! r[s except ! fir/ h be5 respectively add$ to all ! r[s except ! la/4 ,to illu/rate ?is3 let u suppose ! table to be comput$ to be t ( ! fif? p[]s ( ! natural numb]s1 & ! computation to h already proce$$ s f> z ! fif? p[] ( #f1 :i* is #gggf4 ,?is numb] appe>s1 accord+ly1 9 ! highe/ r[1 be+ ! place appropriat$ to ! numb] ( ! table to be calculat$4 ,! sev]al diff]5ces z f> z ! fif?1 :i* is 9 ?is case con/ant1 are exhibit$ on ! successive r[s ( dials 9 su* a mann]1 z to be adapt$ to ! process ( addition by alt]nate r[s1 9 ! mann] already expla9$4 ,! process ( addition w comm5ce by ! motion ( ! dials 9 ! fir/1 ?ird1 & fif? r[s1 9 ! foll[+ mann]3 ,! dial ,a1 fig4 #a1 mu/ turn ?r\gh "o division1 :i* w br+ ! numb] #g to ! index2 ! dial ,b mu/ turn ?r\gh ?ree divisions1 :i* w #j br+ to ! index2 ?is w r5d] a c>riage necess>y1 b t c>riage w n take place dur+ ! pres5t motion ( ! dial4 ,! dial ,c w rema9 unmov$1 s9ce #j is at ! index bel[ x2 ! dial ,d mu/ turn ?r\gh n9e divisions2 & z1 9 do+ s1 ! division betwe5 #i & #j mu/ pass "u ! index1 a c>riage mu/ subsequ5tly take place ^u ! dial to ! left2 ! rema9+ dials ( ! r[ ,t1 fig4 #a1 w rema9 unmov$4 ,9 ! r[ ,d#b ! dial ,a#b w rema9 unmov$1 s9ce #j is at ! index bel[ x2 ! dial ,b#b w be mov$ ?r\gh five divisions1 & w r5d] a subsequ5t c>riage on ! dial to ! left necess>y2 ! dial ,c#b w be mov$ ?r\gh five divisions2 ! dial ,d#b w be mov$ ?r\gh ?ree divisions1 & ! rema9+ dials ( ?is r[ w rema9 unmov$4 ,! dials ( ! r[ ,d#d w be mov$ accord+ to ! same rules2 & ! :ole s*eme w und]go a *ange exhibit$ 9 fig4 #b2 a m>k 7997 be+ introduc$ on ^? dials to :i* a c>riage is r5d]$ necess>y by ! addition :i* has j tak5 place4 ,fig4 #b4 ,! second qu>t] ( a turn ( ! mov+ axis1 w move =w>d ?r\gh "o division all ! dials :i* 9 fig4 #b are m>k$ 79971 & ! s*eme w be conv]t$ into ! s*eme express$ 9 fig4 #c4 ,fig4 #c4 ,9 ! ?ird qu>t] ( a turn1 ! dial ,a#a1 fig4 #c1 w rema9 unmov$1 s9ce #j is at ! index bel[ x2 ! dial ,b#a w be mov$ =w>d ?r\gh ?ree divisions2 ,c#a ?r\gh n9e divisions1 & s on2 & 9 l mann] ! dials ( ! r[ ,d#c w be mov$ =w>d ?r\gh ! numb] ( divisions express$ at ! indices 9 ! r[ ,d#d4 ,?is *ange w conv]t ! arrangem5t into t express$ 9 fig4 #d1 ! dials to :i* a c>riage is due1 be+ di/+ui%$ z be=e by 79974 ,fig4 #d4 ,! f\r? qu>t] ( a turn ( ! axis w move =w>d "o division all ! dials m>k$ 79972 & ! arrangem5t w f9ally assume ! =m exhibit$ 9 fig4 #e1 9 :i* ! calculation is complet$4 ,! fir/ r[ ,t 9 ?is expresses ! fif? p[] ( #g2 & ! second expresses ! numb] :i* mu/ be add$ to ! fir/ r[1 9 ord] to produce ! fif? p[] ( #h2 ! numb]s 9 ea* r[ be+ prep>$ = ! *ange :i* !y mu/ und]go1 9 ord] to enable !m to cont9ue ! computation accord+ to ! me?od ( alt]nate addition "h adopt$4 ,fig4 #e4 ,hav+ ?us expla9$ :at x is t ! me*anism is requir$ to d1 we %all n[ attempt to convey at lea/ a g5]al notion ( "s ( ! me*anical contrivances by :i* ! desir$ ends are atta9$4 ,to simplify ! explanation1 let u fir/ take "o p>ticul> in/ance--! dials ,b & ,b#a1 fig4 #a1 = example4 ,beh9d ! dial ,b#a is a bolt1 :i*1 at ! comm5cem5t ( ! process1 is %ot betwe5 ! tee? ( a :eel :i* drives ! dial ,b3 dur+ ! fir/ qu>t] ( a turn ?is bolt is made to revolve1 & if x cont9u$ to be engag$ 9 ! tee? ( ! said :eel1 x w\ld cause ! dial ,b to make a complete revolution2 b x is necess>y t ! dial ,b %\ld only move ?r\gh ?ree divisions1 &1 !re=e1 :5 ?ree divisions ( ?is dial h pass$ "u its index1 ! a=esaid bolt mu/ be )drawn3 ?is is accompli%$ by a small w$ge1 :i* is plac$ 9 a fix$ position on ! :eel beh9d ! dial ,b#a1 & t position is su* t ?is w$ge w press ^u ! bolt 9 su* a mann]1 t at ! mom5t :5 ?ree divisions ( ! dial ,b h pass$ "u ! index1 x %all )draw ! bolt f ! tee? ( ! :eel :i* x drives4 ,! bolt w cont9ue to revolve dur+ ! rema9d] ( ! fir/ qu>t] ( a turn ( ! axis1 b x w no long] drive ! dial ,b1 :i* w rema9 quiesc5t4 ,_h ! figure at ! index ( ! dial ,b#a be5 any o!r1 ! w$ge :i* )draws ! bolt w\ld h assum$ a diff]5t position1 & w\ld h )drawn ! bolt at a diff]5t "t1 b at a "t always correspond+ ) ! numb] "u ! index ( ! dial ,b#a3 ?us1 if #e _h be5 "u ! index ( ! dial ,b#a1 !n ! bolt w\ld h be5 )drawn f betwe5 ! tee? ( ! :eel :i* x drives1 :5 five divisions ( ! dial ,b _h pass$ "u ! index1 & s on4 ,beh9d ea* dial 9 ! r[ ,d#a !re is a simil> bolt & a simil> )draw+ w$ge1 & ! action ^u ! dial above is transmitt$ & susp5d$ 9 precisely ! same mann]4 ,l obs]vations w be applicable to all ! dials 9 ! s*eme "h ref]r$ to1 9 ref]5ce to _! add+ actions ^u ^? above !m4 ,!re is1 h[ev]1 a p>ticul> case :i* "h m]its notice3 x is ! case 9 :i* #j is "u ! index ( ! dial f :i* ! addition is to be transmitt$ upw>ds4 ,z 9 t case no?+ is to be add$1 a me*anical provision %\ld be made to prev5t ! bolt f engag+ 9 ! tee? ( ! :eel :i* acts ^u ! dial above3 ! w$ge :i* causes ! bolt to be )drawn1 is ?r[n into su* a position z to r5d] x impossible t ! bolt %\ld be %ot1 or t x %\ld ent] betwe5 ! tee? ( ! :eel1 :i* 9 o!r cases x drives4 ,b inasmu* z ! usual means ( %oot+ ! bolt w\ld /ill act1 a /ra9 w\ld necess>ily take place 9 ! p>ts ( ! me*anism1 ow+ to ! bolt n yield+ to ! usual impulse4 ,a small %\ld] is !re=e provid$1 :i* puts aside1 9 ?is case1 ! piece by :i* ! bolt is usually /ruck1 & all[s ! /rik+ implem5t to pass )\t enc\nt]+ ! head ( ! bolt or any o!r ob/ruction4 ,?is me*anism is br\ght into play 9 ! s*eme1 fig4 #a1 9 ! cases ( all ^? dials 9 :i* #j is "u ! index4 ,su* is a g5]al description ( ! nature ( ! me*anism by :i* ! add+ process1 ap>t f ! c>riages1 is effect$4 ,dur+ ! fir/ qu>t] ( a turn1 ! bolts :i* drive ! dials 9 ! fir/1 ?ird1 & fif? r[s1 are caus$ to revolve1 & to act ^u ^! dials1 s long z !y are p]mitt$ by ! position ( ! sev]al w$ges on ! second1 f\r?1 & six? r[s ( dials1 by :i* ^! bolts are respectively )drawn2 &1 dur+ ! ?ird qu>t] ( a turn1 ! bolts :i* drive ! dials ( ! second & f\r? r[s are made to revolve & act ^u ^! dials s long z ! w$ges on ! dials ( ! ?ird & fif? r[s1 :i* )draw !m1 p]mit4 ,x w h5ce be p]ceiv$1 t1 dur+ ! fir/ & ?ird qu>t]s ( a turn1 ! process ( addition is cont9ually pass+ upw>ds ?r\gh ! ma*9]y2 alt]nately f ! ev5 to ! odd r[s1 & f ! odd to ! ev5 r[s1 c\nt+ d[nw>ds4 ,we %all n[ attempt to convey "s notion ( ! me*anism by :i* ! process ( c>ry+ is effect$ dur+ ! second & f\r? qu>t]s ( a turn ( ! axis4 ,z be=e1 we %all fir/ expla9 x 9 ref]5ce to a p>ticul> in/ance4 ,dur+ ! fir/ qu>t] ( a turn ! :eel ,b#b1 fig4 #a1 is caus$ by ! add+ bolt to move ?r\gh five divisions2 & ! fif? ( ^! divisions1 :i* passes "u ! index1 is t betwe5 #i & #j4 ,on ! axis ( ! :eel ,c#b1 imm$iately to ! left ( ,b#b1 is fix$ a :eel1 call$ 9 me*anics a rat*et :eel1 :i* is driv5 by a claw :i* con/antly re/s 9 its tee?4 ,?is claw is 9 su* a position z to p]mit ! :eel ,c#b to move 9 ob$i5ce to ! action ( ! add+ bolt1 b to resi/ its motion 9 ! contr>y direction4 ,x is drawn back by a spiral spr+1 b its recoil is prev5t$ by a hook :i* su/a9s x2 :i* hook1 h[ev]1 is capable ( be+ )drawn1 & :5 )drawn1 ! a=esaid spiral spr+ w\ld draw back ! claw1 & make x fall ?r\gh "o too? ( ! rat*et :eel4 ,n[1 at ! mom5t t ! division betwe5 #i & #j on ! dial ,b#b passes "u ! index1 a ?umb plac$ on ! axis ( ?is dial t\*es a trigg] :i* raises out ( ! not* ! hook :i* su/a9s ! claw j m5tion$1 & all[s x to fall back by ! recoil ( ! spr+1 & to drop into ! next too? ( ! rat*et :eel4 ,?is process1 h[ev]1 produces no imm$iate effect ^u ! position ( ! :eel ,c#b1 & is m]ely prep>atory to an action int5d$ to take place dur+ ! second qu>t] ( a turn ( ! mov+ axis4 ,x is 9 effect a memor&um tak5 by ! ma*9e ( a c>riage to be made 9 ! next qu>t] ( a turn4 ,dur+ ! second qu>t] ( a turn1 a f+] plac$ on ! axis ( ! dial ,b#b is made to revolve1 & x enc\nt]s ! heel ( ! above-m5tion$ claw4 ,z x moves =w>d x drives ! claw be=e x3 & ?is claw1 re/+ 9 ! tee? ( ! rat*et :eel fix$ ^u ! axis ( ! dial ,c#b drives =w>d t :eel1 & ) x ! dial4 ,b ! l5g? & position ( ! f+] :i* drives ! claw limits its action1 s z to move ! claw =w>d ?r\gh su* a space only z w cause ! dial ,c#b to advance ?r\gh a s+le division2 at :i* po9t x is aga9 caught & reta9$ by ! hook4 ,?is w be add$ to ! numb] "u its index1 & ! requisite c>riage f ,b#b to ,c#b w be accompli%$4 ,9 connexion ) e dial is plac$ a simil> rat*et :eel ) a simil> claw1 drawn by a simil> spr+1 su/a9$ by a simil> hook1 & act$ ^u by a simil> ?umb & trigg]2 & !re=e ! necess>y c>riages1 ?r\gh\t ! :ole ma*9]y1 take place 9 ! same mann] & by simil> means4 ,dur+ ! second qu>t] ( a turn1 su* ( ! c>ry+ claws z h be5 all[$ to recoil 9 ! fir/1 ?ird1 & fif? r[s1 are drawn up by ! f+]s on ! axes ( ! adjac5t dials2 &1 dur+ ! f\r? qu>t] ( a turn1 su* ( ! c>ry+ claws on ! second & f\r? r[s z h be5 all[$ to recoil dur+ ! ?ird qu>t] ( a turn1 are 9 l mann] drawn up by ! c>ry+ f+]s on ! axes ( ! adjac5t dials4 ,x appe>s t ! c>riages proce$ alt]nately f "r to left along ! horizontal r[s dur+ ! second & f\r? qu>t]s ( a turn2 9 ! "o1 !y pass along ! fir/1 ?ird1 & fif? r[s1 & 9 ! o!r1 along ! second & f\r?4 ,!re are two sy/ems ( waves ( me*anical action cont9ually fl[+ f ! bottom to ! top2 & two /reams ( simil> action con/antly pass+ f ! "r to ! left4 ,! cre/s ( ! fir/ sy/em ( add+ waves fall ^u ! la/ diff]5ce1 & ^u e alt]nate "o proce$+ upw>ds2 :ile ! cre/s ( ! o!r sy/em t\* ^u ! int]m$iate diff]5ces4 ,! fir/ /ream ( c>ry+ action passes f "r to left along ! highe/ r[ & e alt]nate r[1 :ile ! second /ream passes along ! int]m$iate r[s4 ,su* is a v rapid & g5]al outl9e ( ?is ma*9]y4 ,its wond]s1 h[ev]1 are /ill great] 9 its details ?an ev5 9 its broad] features4 ,al?\gh we despair ( do+ x ju/ice by any description :i* c be attempt$ "h1 yet we %\ld n fulfil ! duty we owe to our read]s1 if we did n call _! att5tion at lea/ to a few ( ! in/ances ( consummate skill :i* are scatt]$1 ) a prodigality *>act]i/ic ( ! highe/ ord] ( inv5tive g5ius1 ?r\gh\t ?is a/oni%+ me*anism4 ,9 ! g5]al description :i* we h giv5 ( ! me*anism = c>ry+1 x w be obs]v$1 t ! prep>ation = e c>riage is /at$ to be made dur+ ! previ\s addition1 by ! dis5gagem5t ( ! c>ry+ claw be=e m5tion$1 & by its consequ5t recoil1 urg$ by ! spiral spr+ ) :i* x is connect$2 b x may1 & does1 frequ5tly happ51 t ?\gh ! process ( addition may n h r5d]$ a c>riage necess>y1 "o c>riage may itself produce ! necessity = ano!r4 ,?is is a cont+5cy n provid$ aga9/ 9 ! me*anism z we h describ$ x3 ! case w\ld occur 9 ! s*eme repres5t$ 9 fig4 #a1 if ! figure "u ! index ( ,c#b 7 #d in/ead ( #c4 ,! addition ( ! numb] #e at ! index ( ,c#c w\ld1 9 ?is case1 9 ! fir/ qu>t] ( a turn1 br+ #i to ! index ( ,c#b3 ?is w\ld obvi\sly r5d] no c>riage necess>y1 & ( c\rse no prep>ation w\ld be made = "o by ! me*anism--t is to say1 ! c>ry+ claw ( ! :eel ,d#b w\ld n be deta*$4 ,mean:ile a c>riage ^u ,c#b has be5 r5d]$ necess>y by ! addition made 9 ! fir/ qu>t] ( a turn to ,b#b4 ,?is c>riage takes place 9 ! ord9>y way1 & w\ld cause ! dial ,c#b1 9 ! second qu>t] ( a turn1 to advance f #i to #j3 ?is w\ld make ! necess>y prep>ation = a c>riage f ,c#b to ,d#b4 ,b unless "s special arrangem5t 0 made = ! purpose1 t c>riage w\ld n take place dur+ ! second qu>t] ( a turn4 ,?is peculi> cont+5cy is provid$ aga9/ by an arrangem5t ( s+ul> me*anical beauty1 & :i*1 at ! same "t1 answ]s ano!r purpose--t ( equaliz+ ! resi/ance oppos$ to ! mov+ p[] by ! c>ry+ me*anism4 ,! f+]s plac$ on ! axes ( ! sev]al dials 9 ! r[ ,d#b1 d n act at ! same in/ant on ! c>ry+ claws adjac5t to !m2 b !y are s plac$1 t _! action may be di/ribut$ ?r\gh\t ! second qu>t] ( a turn 9 regul> succession4 ,?us ! f+] on ! axis ( ! dial ,a#b fir/ enc\nt]s ! claw ^u ,b#b1 & drives x ?r\gh "o too? imm$iately =w>ds2 ! f+] on ! axis ( ,b#b enc\nt]s ! claw ^u ,c#b & drives x ?r\gh "o too?2 ! action ( ! f+] on ,c#b on ! claw on ,d#b next succe$s1 & s on4 ,?us1 :ile ! f+] on ,b#b acts on ,c#b1 & causes ! division f #i to #j to pass "u ! index1 ! ?umb on ,c#b at ! same in/ant acts on ! trigg]1 & deta*es ! c>ry+ claw on ,d#b1 :i* is =?) enc\nt]$ by ! c>ry+ f+] on ,c#b1 & driv5 =w>d "o too?4 ,! dial ,d#b accord+ly moves =w>d "o division1 & #e is br\ght "u ! index4 ,?is arrangem5t is beautifully effect$ by plac+ ! sev]al f+]s1 :i* act ^u ! c>ry+ claws1 spirally on _! axes1 s t !y come into action 9 regul> succession4 ,we h /at$ t1 at ! comm5cem5t ( ea* revolution ( ! mov+ axis1 ! bolts :i* drive ! dials ( ! fir/1 ?ird1 & fif? r[s1 are %ot4 ,! process ( %oot+ ^! bolts mu/ !re=e h tak5 place dur+ ! la/ qu>t] ( ! prec$+ revolution2 b x is dur+ t qu>t] ( a turn t ! c>riages are effect$ 9 ! second & f\r? r[s4 ,s9ce ! bolts :i* drive ! dials ( ! fir/1 ?ird1 & fif? r[s1 h no me*anical connexion ) ! dials 9 ! second & f\r? r[s1 !re is no?+ 9 ! process ( %oot+ ^? bolts incompatible ) t ( mov+ ! dials ( ! second & f\r? r[s3 h5ce ^! two processes may bo? take place dur+ ! same qu>t] ( a turn4 ,b 9 ord] to equalize ! resi/ance to ! mov+ p[]1 ! same exp$i5t is "h adopt$ z t already describ$ 9 ! process ( c>ry+4 ,! arms :i* %oot ! bolts ( ea* r[ ( dials are arrang$ spirally1 s z to act successively ?r\gh\t ! qu>t] ( a turn4 ,!re is1 h[ev]1 a cont+5cy :i*1 "u c]ta9 circum/ances1 w\ld "h produce a difficulty :i* mu/ be provid$ aga9/4 ,x is possible1 & 9 fact does sometimes happ51 t ! process ( c>ry+ causes a dial to move "u ! index f #j to #a4 ,9 t case1 ! bolt1 prep>atory to ! next addition1 ought n to be %ot until aft] ! c>riage takes place2 = if ! arm :i* %oots x passes its po9t ( action be=e ! c>riage takes place1 ! bolt w be mov$ out ( its sph]e ( action1 & w n be %ot1 :i*1 z we h already expla9$1 mu/ always happ5 :5 #j is at ! index3 !re=e no addition w\ld 9 ?is case take place dur+ ! next qu>t] ( a turn ( ! axis2 :]eas1 s9ce #a is br\ght to ! index by ! c>riage1 :i* imm$iately succe$s ! passage ( ! arm :i* ought to bolt1 #a %\ld be add$ dur+ ! next qu>t] ( a turn4 ,x is pla91 accord+ly1 t ! me*anism %\ld be s arrang$1 t ! action ( ! arms1 :i* %oot ! bolts successively1 %\ld imm$iately foll[ ! action ( ^? f+]s :i* raise ! c>ry+ claws successively2 & !re=e ei!r a sep>ate qu>t] ( a turn %\ld be appropriat$ to ea* ( ^? movem5ts1 or if !y be execut$ 9 ! same qu>t] ( a turn1 ! me*anism mu/ be s con/ruct$1 t ! arms :i* %oot ! bolts successively1 %all sev]ally foll[ imm$iately aft] ^? :i* raise ! c>ry+ claws successively4 ,! latt] object is atta9$ by a me*anical arrangem5t ( s+ul> felicity1 & p>tak+ ( t elegance :i* *>act]ises all ! details ( ?is me*anism4 ,bo? sets ( arms are spirally arrang$ on _! respective axes1 s z to be c>ri$ ?r\gh _! p]iod 9 ! same qu>t] ( a turn2 b ! "o spiral is %ift$ a few degrees1 9 angul> position1 beh9d ! o!r1 s t ea* pair ( correspond+ arms succe$ ea* o!r 9 ! mo/ regul> ord]1--equaliz+ ! resi/ance1 economiz+ "t1 h>moniz+ ! me*anism1 & giv+ to ! :ole me*anical action ! utmo/ practical p]fection4 ,! sy/em ( me*anical contrivances by :i* ! results1 "h attempt$ to be describ$1 are atta9$1 =m only "o ord] ( exp$i5ts adopt$ 9 ?is ma*9]y2--al?\gh su* is ! p]fection ( _! action1 t 9 any ord9>y case !y w\ld be reg>d$ z hav+ atta9$ ! ends 9 view ) an almo/ sup]flu\s degree ( precision4 ,consid]+1 h[ev]1 ! imm5se importance ( ! purposes :i* ! me*anism 0 de/9$ to fulfil1 its inv5tor det]m9$ t a high] ord] ( exp$i5ts %\ld be sup]9duc$ ^u ^? already describ$2 ! purpose ( :i* %\ld be to oblit]ate all small errors or inequalities :i* might1 ev5 by remote possibility1 arise1 ei!r f defects 9 ! orig9al =mation ( ! me*anism1 f inequality ( we>1 f casual /ra9 or d]angem5t1--or1 9 %ort1 f any o!r cause :atev]4 ,?us ! movem5ts ( ! fir/ & pr9cipal p>ts ( ! me*anism 7 reg>d$ by him m]ely z a fir/1 ?\gh extremely nice approximation1 ^u :i* a sy/em ( small corrections 0 to be subsequ5tly made by suitable & indep5d5t me*anism4 ,?is supplem5t>y sy/em ( me*anism is s contriv$1 t if "o or m ( ! mov+ p>ts ( ! me*anism ( ! fir/ ord] be slightly out ( _! places1 !y w be =c$ to _! exact position by ! action ( ! me*anical exp$i5ts ( ! second ord] to :i* we n[ allude4 ,if a m consid]able d]angem5t 7 produc$ by any accid5tal di/urbance1 ! consequ5ce w\ld be t ! supplem5t>y me*anism w\ld cause ! :ole sy/em to become lock$1 s t n a :eel w\ld be capable ( mov+2 ! impell+ p[] w\ld necess>ily lose all its en]gy1 & ! ma*9e w\ld /op4 ,! consequ5ce ( ?is exquisite arrangem5t is1 t ! ma*9e w ei!r calculate rightly1 or n at all4 ,! sup]num]>y contrivances :i* we n[ allude to1 be+ 9 a great degree unconnect$ ) ea* o!r1 & scatt]$ ?r\gh ! ma*9]y to a c]ta9 ext5t1 indep5d5t ( ! me*anical arrangem5t ( ! pr9cipal p>ts1 we f9d x difficult to convey any di/9ct notion ( _! nature or =m4 ,9 "s in/ances !y consi/ ( a roll] re/+ betwe5 c]ta9 curv$ surfaces1 :i* has b "o position ( /able equilibrium1 & t position ! same1 h[ev] ! roll] or ! curv$ surfaces may we>4 ,a slight error 9 ! motion ( ! pr9cipal p>ts w\ld make ?is roll] = ! mom5t re/ on "o ( ! curves2 b1 be+ con/antly urg$ by a spr+1 x w\ld press on ! curv$ surface 9 su* a mann] z to =ce ! mov+ piece on :i* t curv$ surface is =m$1 into su* a position t ! roll] may re/ betwe5 ! two surfaces2 t position be+ ! "o :i* ! me*anism %\ld h4 ,a great] d]angem5t w\ld br+ ! roll] to ! cre/ ( ! curve1 on :i* x w\ld re/ 9 in/able equilibrium2 & ! ma*9e w\ld ei!r become lock$1 or ! roll] w\ld ?r[ x z be=e into its true position4 ,9 o!r in/ances a simil> object is atta9$ by a solid cone be+ press$ into a conical seat2 ! position ( ! axis ( ! cone & t ( its seat be+ necess>ily inv>iable1 h[ev] ! cone may we>3 & ! action ( ! cone ^u ! seat be+ su*1 t x _c re/ 9 any position except t 9 :i* ! axis ( ! cone co9cides ) ! axis ( its seat4 ,hav+ ?us attempt$ to convey a notion1 h[ev] inadequate1 ( ! calculat+ section ( ! ma*9]y1 we %all proce$ to (f] "s explanation ( ! means :]eby x is enabl$1 to pr9t its calculations 9 su* a mann] z to preclude ! possibility ( error 9 any individual pr9t$ copy4 ,on ! axle ( ea* ( ! :eels :i* express ! calculat$ numb] ( ! table ,t1 !re is fix$ a solid piece ( metal1 =m$ into a curve1 n unlike ! :eel 9 a common clock1 :i* is call$ ! snail4 ,?is curv$ surface acts aga9/ ! arm ( a lev]1 s z to raise t arm to a high] or l[] po9t accord+ to ! position ( ! dial ) :i* ! snail is connect$4 ,)\t ent]+ into a m m9ute description1 x w be easily und]/ood t ! snail may be s =m$ t ! arm ( ! lev] %all be rais$ to t5 diff]5t elevations1 correspond+ to ! t5 figures ( ! dial :i* may be br\ght "u ! index4 ,! opposite arm ( ! lev] "h describ$ puts 9 motion a solid ar*1 or sector1 :i* c>ries t5 pun*es3 ea* pun* be>+ on its face a rais$ "* ( a figure1 & ! t5 pun*es be>+ ! t5 *>act]s1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #j4 ,x w be app>5t f :at has be5 j /at$1 t ?is type sector 7z x is call$7 w receive t5 diff]5t attitudes1 correspond+ to ! t5 figures :i* may successively be br\ght "u ! index ( ! dial-plate4 ,at a po9t ov] :i* ! type sector is ?us mov$1 & imm$iately "u a po9t ?r\gh :i* x plays1 is plac$ a frame1 9 :i* is fix$ a plate ( copp]4 ,imm$iately ov] a c]ta9 po9t ?r\gh :i* ! type sector moves1 is likewise plac$ a b5t lev]1 :i*1 be+ /raight5$1 is =cibly press$ ^u ! pun* :i* has be5 br\ght "u x4 ,if ! type sector be mov$1 s z to br+ "u ! b5t lev] "o ( ! /eel pun*es above m5tion$1 & be held 9 t position = a c]ta9 "t1 ! b5t lev]1 be+ /raight5$1 acts ^u ! /eel pun*1 & drives x aga9/ ! face ( ! copp] b5ea?1 & ?us causes a sunk5 impression ( ! "* ^u ! pun* to be left ^u ! copp]4 ,if ! copp] be n[ %ift$ slightly 9 its position1 & ! type sector be also %ift$ s z to br+ ano!r pun* "u ! b5t lev]1 ano!r "* may be engrav$ on ! copp] by /raight5+ ! b5t lev]1 & press+ x on ! pun* z be=e4 ,x w be evid5t1 t if ! copp] 0 %ift$ f "r to left ?r\gh a space equal to two figures ( a numb]1 &1 at ! same "t1 ! type sector s %ift$ z to br+ ! pun*es correspond+ to ! figures ( ! numb] successively "u ! b5t lev]1 an engrav$ impression ( ! numb] might ?us be obta9$ ^u ! copp] by ! cont9u$ action ( ! b5t lev]4 ,if1 :5 "o l9e ( figures is ?us obta9$1 a provision be made to %ift ! copp] 9 a direction at "r angles to its =m] motion1 ?r\gh a space equal to ! di/ance betwe5 two l9es ( figures1 & at ! same "t to %ift x ?r\gh a space 9 ! o!r direction equal to ! l5g? ( an entire l9e1 x w be evid5t t ano!r l9e ( figures might be pr9t$ bel[ ! fir/ 9 ! same mann]4 ,! motion ( ! type sector1 "h describ$1 is accompli%$ by ! action ( ! snail ^u ! lev] already m5tion$4 ,9 ! case :]e ! numb] calculat$ is t express$ 9 fig4 #a1 ! process w\ld be z foll[s3--,! snail ( ! :eel ,f#a1 act+ ^u ! lev]1 w\ld ?r[ ! type sector into su* an attitude1 t ! pun* be>+ ! "* #j w\ld come "u ! b5t lev]4 ,! next turn ( ! mov+ axis w\ld cause ! b5t lev] to press on ! tail ( ! pun*1 & ! "* #j w\ld be impress$ ^u ! copp]4 ,! b5t lev] be+ aga9 drawn up1 ! pun* w\ld recoil f ! copp] by ! action ( a spr+2 ! next turn ( ! mov+ axis w\ld %ift ! copp] ?r\gh ! int]val betwe5 two figures1 s z to br+ ! po9t de/9$ to be impress$ ) ! next figure "u ! b5t lev]4 ,at ! same "t1 ! snail ( ! :eel ,e w\ld cause ! type sector to be ?r[n into ! same attitude z be=e1 & ! pun* w\ld be br\ght "u ! b5t lev]2 ! next turn w\ld impress ! figure beside ! =m] "o1 z be=e describ$4 ,! snail ^u ! :eel ,d w\ld n[ come into action1 & ?r[ ! type sector into t position 9 :i* ! pun* be>+ ! "* #g w\ld come "u ! b5t lev]1 & at ! same "t ! copp] w\ld be %ift$ ?r\gh ! int]val betwe5 two figures2 ! /raight5+ ( ! lev] w\ld next foll[1 & ! "* #g w\ld be engrav$4 ,9 ! same mann]1 ! :eels ,c1 ,b1 & ,a w\ld successively act by means ( _! snails2 & ! copp] be+ %ift$1 & ! lev] all[$ to act1 ! numb] #jjgggf w\ld be f9ally engrav$ ^u ! copp]3 ?is be+ accompli%$1 ! calculat+ ma*9]y w\ld next be call$ into action1 & ano!r calculation w\ld be made1 produc+ ! next numb] ( ! ,table exhibit$ 9 fig4 #e4 ,dur+ ?is process ! ma*9]y w\ld be engag$ 9 %ift+ ! copp] bo? 9 ! direction ( its l5g? & its bread?1 ) a view to comm5ce ! pr9t+ ( ano!r l9e2 & ?is *ange ( position w\ld be accompli%$ at ! mom5t :5 ! next calculation w\ld be complet$3 ! pr9t+ ( ! next numb] w\ld g on l ! =m]1 & ! op]ation ( ! ma*9e w\ld proce$ 9 ! same mann]1 calculat+ & pr9t+ alt]nately4 ,x is n1 h[ev]1 at all necess>y--?\gh we h "h suppos$ x1 = ! sake ( simplify+ ! explanation--t ! calculat+ "p ( ! me*anism %\ld h its action susp5d$ :ile ! pr9t+ "p is 9 op]ation1 or vice v]sa2 x is n int5d$1 9 fact1 to be s susp5d$ 9 ! actual ma*9]y4 ,! same turn ( ! axis by :i* "o numb] is pr9t$1 executes a "p ( ! movem5ts necess>y = ! succe$+ calculation2 s t ! :ole me*anism w be simultane\sly & cont9u\sly 9 action4 ,( ! me*anism by :i* ! position ( ! copp] is %ift$ f figure to figure1 f l9e to l9e1 we %all n attempt any description4 ,we feel t x w\ld be q va94 ,complicat$ & difficult to describe z e o!r "p ( ?is ma*9]y is1 ! me*anism = mov+ ! copp] is su* z x w\ld be q impossible to r5d] at all intelligible1 )\t num]\s illu/rative draw+s4 ,! engrav$ plate ( copp] obta9$ 9 ! mann] above describ$1 is design$ to be us$ z a m\ld f :i* a /]eotyp$ plate may be ca/2 or1 if deem$ advisable1 x may be us$ z ! imm$iate means ( pr9t+4 ,9 ! "o case we %\ld produce a table1 pr9t$ f type1 9 ! same mann] z common lett]-press pr9t+2 9 ! o!r an engrav$ table4 ,if x be ?\ght mo/ advisable to pr9t f ! /]eotyp$ plates1 !n z _m /]eotyp$ plates z may be requir$ may be tak5 f ! copp] m\ld2 s t :5 once a table has be5 calculat$ & engrav$ by ! ma*9]y1 ! :ole _w may be suppli$ ) /]eotyp$ plates to pr9t x1 & may cont9ue to be s suppli$ = an unlimit$ p]iod ( "t4 ,!re is no practical limit to ! numb] ( /]eotyp$ plates :i* may be tak5 f ! engrav$ copp]2 & !re is sc>cely any limit to ! numb] ( pr9t$ copies :i* may be tak5 f any s+le /]eotyp$ plate4 ,n only1 !re=e1 is ! num]ical table by ^! means engrav$ & /]eotyp$ ) infallible accuracy1 b su* /]eotyp$ plates are producible 9 unb\nd$ quantity4 ,ea* plate1 :5 produc$1 becomes itself ! means ( produc+ pr9t$ copies ( ! table1 9 accuracy p]fect1 & 9 numb] )\t limit4 ,unlike all o!r ma*9]y1 ! calculat+ me*anism produces1 n ! object ( consumption1 b ! ma*9]y by :i* t object may be made4 ,to say t x computes & pr9ts ) infallible accuracy1 is to und]/ate its m]its3--x computes & fabricates ! means ( pr9t+ ) absolute correctness & 9 unlimit$ abundance4 ,= ! sake ( cle>ness1 & to r5d] ourselves m easily intelligible to ! g5]al read]1 we h 9 ! prec$+ explanation ?r[n ! me*anism into an arrangem5t some:at diff]5t f t :i* is really adopt$4 ,! dials express+ ! numb]s ( ! tables ( ! successive diff]5ces are n plac$1 z we h suppos$ !m1 9 horizontal r[s1 & read f "r to left1 9 ! ord9>y way2 !y are1 on ! contr>y1 plac$ v]tically1 "o bel[ ! o!r1 & read f top to bottom4 ,! numb] ( ! table occupies ! fir/ v]tical column on ! "r1 ! units be+ express$ on ! l[e/ dial1 & ! t5s on ! next above t1 & s on4 ,! fir/ diff]5ce occupies ! next v]tical column on ! left2 & ! numb]s ( ! succe$+ diff]5ces occupy v]tical columns1 proce$+ regul>ly to ! left2 ! con/ant diff]5ce be+ on ! la/ v]tical column4 ,x is int5d$ 9 ! ma*9e n[ 9 progress to introduce six ord]s ( diff]5ces1 s t !re w be sev5 columns ( dials2 x is also int5d$ t ! calculations %all ext5d to eighte5 places ( figures3 ?us ea* column w h eighte5 dials4 ,we h ref]r$ to ! dials z if !y 7 inscrib$ ^u ! faces ( :eels1 ^: axes are horizontal & planes v]tical4 ,9 ! actual ma*9]y ! axes are v]tical & ! planes horizontal1 s t ! edges ( ! figure :eels1 z !y are call$1 are pres5t$ to ! eye4 ,! figures are inscrib$1 n ^u ! dial-plate1 b ar\nd ! surface ( a small cyl9d] or b>rel1 plac$ ^u ! axis ( ! figure :eel1 :i* revolves ) x2 s t z ! figure :eel revolves1 ! figures on ! b>rel are successively br\ght to ! front1 & pass "u an index engrav$ ^u a plate ( metal imm$iately above ! b>rel4 ,?is arrangem5t has ! obvi\s practical advantage1 t1 in/ead ( ea* figure :eel hav+ a sep>ate axis1 all ! figure :eels ( ! same v]tical column revolve on ! same axis2 & ! same obs]vation w apply to all ! :eels ) :i* ! figure :eels are 9 me*anical connexion4 ,?is arrangem5t has ! fur!r me*anical advantage ov] t :i* has be5 assum$ = ! purposes ( explanation1 t ! friction ( ! :eel-"w on ! axes is less 9 am\nt1 & m uni=mly di/ribut$1 ?an x c\ld be if ! axes 7 plac$ 9 ! horizontal position4 ,a notion may !re=e be =m$ ( ! front elevation ( ! calculat+ "p ( ! me*anism1 by conceiv+ sev5 /eel axes erect$1 "o beside ano!r1 on ea* ( :i* %all be plac$ eighte5 :eels1#ab five in*es 9 diamet]1 hav+ cyl9d]s or b>rels ^u !m an in* & a half 9 height1 & inscrib$1 z already /at$1 ) ! t5 ari?metical *>act]s4 ,! entire elevation ( ! ma*9]y w\ld occupy a space measur+ t5 feet broad1 t5 feet high1 & five feet deep4 ,! process ( calculation w\ld be obs]v$ by ! alt]nate motion ( ! figure :eels on ! sev]al axes4 ,dur+ ! fir/ qu>t] ( a turn1 ! :eels on ! fir/1 ?ird1 & fif? axes w\ld turn1 receiv+ _! addition f ! second1 f\r?1 & six?2 dur+ ! second qu>t] ( a turn1 su* ( ! :eels on ! fir/1 ?ird1 & fif? axes1 to :i* c>riages are due1 w\ld be mov$ =w>d "o additional figure2 ! second1 f\r?1 & six? columns ( :eels be+ all ?is "t quiesc5t4 ,dur+ ! ?ird qu>t] ( a turn1 ! second1 f\r?1 & six? columns w\ld be obs]v$ to move1 receiv+ _! additions f ! ?ird1 fif?1 & sev5? axes2 & dur+ ! f\r? qu>t] ( a turn1 su* ( ^! :eels to :i* c>riages are due1 w\ld be obs]v$ to move =w>d "o additional figure2 ! :eels ( ! fir/1 ?ird1 & fif? columns be+ quiesc5t dur+ ?is "t4 ,footnote #ab3 ,! :eels1 & e o!r "p ( ! me*anism except ! axes1 spr+s1 & su* p>ts z are necess>ily ( /eel1 are =m$ ( an alloy ( copp] ) a small portion ( t94 ,x w be obs]v$ t ! :eels ( ! sev5? column are always quiesc5t 9 ?is process2 & x may be ask$1 ( :at use !y are1 & :e!r "s me*anism ( a fix$ nature w\ld n s]ve ! same purpose8 ,x mu/1 h[ev]1 be rememb]$1 t = diff]5t tables !re w be diff]5t con/ant diff]5ces2 & t :5 ! calculation ( a table is ab\t to comm5ce1 ! :eels on ! sev5? axis mu/ be mov$ by ! h&1 s z to express ! con/ant diff]5ce1 :atev] x may be4 ,9 tables1 also1 :i* h n a diff]5ce rigor\sly con/ant1 x w be necess>y1 aft] a c]ta9 numb] ( calculations1 to *ange ! con/ant diff]5ce by ! h&2 & 9 ?is case ! :eels ( ! sev5? axis mu/ be mov$ :5 occasion requires4 ,su* adju/m5t1 h[ev]1 w only be necess>y at v di/ant int]vals1 & aft] a consid]able ext5t ( pr9t+ & calculation has tak5 place2 & :5 x is necess>y1 a provision is made 9 ! ma*9]y by :i* notice w be giv5 by ! s\nd+ ( a bell1 s t ! ma*9e may n run beyond ! ext5t ( its p[]s ( calculation4 ,imm$iately beh9d ! sev5 axes on :i* ! figure :eels revolve1 are sev5 o!r axes2 on :i* are plac$1 fir/1 ! :eels already describ$ z driv5 by ! figure :eels1 & :i* be> ^u !m ! w$ge :i* )draws ! bolt imm$iately ov] ^! latt] :eels1 & on ! same axis is plac$ ! add+ bolt4 ,f ! bottom ( ?is bolt !re projects d[nw>ds ! p91 :i* acts ^u ! unbolt+ w$ge by :i* ! bolt is )drawn3 f ! upp] surface ( ! bolt proce$s a too?1 :i*1 :5 ! bolt is %ot1 ent]s betwe5 ! tee? ( ! add+ :eel1 :i* turns on ! same axis1 & is plac$ imm$iately above ! bolt3 its tee?1 on :i* ! bolt acts1 are l ! tee? ( a cr[n :eel1 & are pres5t$ d[nw>ds4 ,! bolt is fix$ ^u ?is axis1 & turns ) x2 b ! add+ :eel above ! bolt1 & ! unbolt+ :eel bel[ x1 bo? turn ^u ! axis1 & indep5d5tly ( x4 ,:5 ! axis is made to revolve by ! mov+ p[]1 ! bolt revolves ) x2 & s long z ! too? ( ! bolt rema9s ins]t$ betwe5 ^? ( ! add+ :eel1 ! latt] is likewise mov$2 b :5 ! l[] p9 ( ! bolt enc\nt]s ! unbolt+ w$ge on ! l[] :eel1 ! too? ( ! bolt is )drawn1 & ! motion ( ! add+ :eel is /opp$4 ,?is add+ :eel is furni%$ ) spur tee?1 besides ! cr[n tee? j m5tion$2 & ^! spur tee? are engag$ ) ^? ( t unbolt+ :eel :i* is 9 connexion ) ! adjac5t figure :eel to :i* ! addition is to be made4 ,by su* an arrangem5t x is evid5t t ! revolution ( ! bolt w necess>ily add to ! adjac5t figure :eel ! requisite numb]4 ,x w be p]ceiv$1 t ^u ! same axis are plac$ an unbolt+ :eel1 a bolt1 & an add+ :eel1 "o above ! o!r1 = e figure :eel2 & z !re are eighte5 figure :eels !re w be eighte5 ti]s2 ea* ti] =m$ ( an unbolt+ :eel1 a bolt1 & an add+ :eel1 plac$ "o above ! o!r2 ! :eels on ?is axis all revolv+ indep5d5t ( ! axis1 b ! bolts be+ all fix$ ^u x4 ,! same obs]vations1 ( c\rse1 w apply to ea* ( ! sev5 axes4 ,at ! comm5cem5t ( e revolution ( ! add+ axes1 x is evid5t t ! sev]al bolts plac$ ^u !m mu/ be %ot 9 ord] to p]=m ! v>i\s additions4 ,?is is accompli%$ by a ?ird set ( sev5 axes1 plac$ at "s di/ance beh9d ! range ( ! :eels1 :i* turn ^u ! add+ axes3 ^! are call$ bolt+ axes4 ,on ^! bolt+ axes are fix$1 s z to revolve ) !m1 a bolt+ f+] opposite to ea* bolt2 z ! bolt+ axis is made to revolve by ! mov+ p[]1 ! bolt+ f+] is turn$1 & z x passes ne> ! bolt1 x enc\nt]s ! %\ld] ( a hamm] or lev]1 :i* /rikes ! heel ( ! bolt1 & presses x =w>d s z to %oot its too? betwe5 ! cr[n tee? ( ! add+ :eel4 ,! only exception to ?is action is ! case 9 :i* #j happ5s to be at ! index ( ! figure :eel2 9 t case1 ! lev] or hamm]1 :i* ! bolt+ f+] w\ld enc\nt]1 is1 z be=e /at$1 lift$ out ( ! way ( ! bolt+ f+]1 s t x revolves )\t enc\nt]+ x4 ,x is on ! bolt+ axes t ! f+]s are spirally arrang$ s z to equalize _! action1 z already expla9$4 ,! same axes 9 ! front ( ! ma*9]y on :i* ! figure :eels turn1 are made to s]ve ! purpose ( c>ry+4 ,ea* ( ^! be> a s]ies ( f+]s :i* turn ) !m1 & :i* enc\nt] a c>ry+ claw1 already describ$1 s z to make ! c>riage3 ^! c>ry+ f+]s are also spirally arrang$ on _! axes1 z already describ$4 ,al?\gh ! absolute accuracy :i* appe>s to be ensur$ by ! me*anical arrangem5ts "h describ$ is su* z to r5d] fur!r precautions ne>ly sup]flu\s1 /ill x may be "r to /ate1 t1 suppos+ x 7 possible = an error to be produc$ 9 calculation1 ?is error c\ld be easily & spe$ily detect$ 9 ! pr9t$ tables3 x w\ld only be necess>y to calculate a numb] ( ! table tak5 at int]vals1 ?r\gh :i* ! me*anical action ( ! ma*9e has n be5 susp5d$1 & dur+ :i* x has receiv$ no adju/m5t by ! h&3 if ! comput$ numb] be f\nd to agree ) ^? pr9t$1 x may be tak5 = grant$ t all ! int]m$iate numb]s are correct2 because1 f ! nature ( ! me*anism1 & ! pr9ciple ( computation1 an error occurr+ 9 any s+le numb] ( ! table w\ld be unavoidably entail$1 9 an increas+ ratio1 ^u all ! succe$+ numb]s4 ,we h hi!rto spok5 m]ely ( ! practicability ( execut+ by ! ma*9]y1 :5 complet$1 t :i* its inv5tor orig9ally contemplat$--namely1 ! calculat+ & pr9t+ ( all num]ical tables1 d]iv$ by ! me?od ( diff]5ces f a con/ant diff]5ce4 ,x has1 h[ev]1 happ5$ t ! actual p[]s ( ! ma*9]y greatly transc5d ^? contemplat$ 9 its orig9al design3--!y n only h exce$$ ! mo/ sangu9e anticipations ( its inv5tor1 b !y appe> to h an ext5t to :i* x is utt]ly impossible1 ev5 = ! mo/ acute ma!matical ?9k]1 to fix a probable limit4 ,c]ta9 subsidi>y me*anical inv5tions h1 9 ! progress ( ! ent]prise1 be51 by ! v nature ( ! ma*9]y1 sugge/$ to ! m9d ( ! inv5tor1 :i* conf] ^u x capabilities :i* he _h nev] =ese54 ,x w\ld be impossible ev5 to enum]ate1 )9 ! limits ( ?is article1 mu* less to describe 9 detail1 ^? extraord9>y me*anical arrangem5ts1 ! effects ( :i* h n fail$ to /rike ) a/oni%m5t e "o :o has be5 fav\r$ ) an opportunity ( witness+ !m1 & :o has be5 enabl$1 by suffici5t ma!matical atta9m5ts1 9 any degree to e/imate _! probable consequ5ces4 ,z we h describ$ ! me*anism1 ! axes conta9+ ! sev]al diff]5ces are successively & regul>ly add$ "o to ano!r2 b !re are c]ta9 me*anical adju/m5ts1 & ^! ( a v simple nature1 :i* be+ ?r[n into action1 w cause a diff]5ce ( any ord] to be add$ any numb] ( times to a diff]5ce ( any o!r ord]2 & t ei!r proce$+ backw>ds or =w>ds1 f a diff]5ce ( an inf]ior to "o ( a sup]ior ord]1 & vice v]sa4#ac ,footnote #ac3 ,! ma*9e 0 con/ruct$ ) ! int5tion ( tabulat+ ! equation ,delta#gu #j1 b1 by ! means above allud$ to1 x is capable ( tabulat+ su* equations z ! foll[+3 ,delta#gu a ,delta u1 ,delta#gu a,delta#cu1 ,delta#gu units figure ( ,delta u4 ,among o!r peculi> me*anical provisions 9 ! ma*9]y is "o by :i*1 :5 ! table = any ord] ( diff]5ce am\nts to a c]ta9 numb]1 a c]ta9 ari?metical *ange w\ld be made 9 ! con/ant diff]5ce4 ,9 ?is way a s]ies may be tabulat$ by ! ma*9e1 9 :i* ! con/ant diff]5ce is subject to p]iodical *ange2 or ! v nature ( ! table itself may be subject to p]iodical *ange1 & yet to "o :i* has a regul> law4 ,"s ( ^! subsidi>y p[]s are peculi>ly applicable to calculations requir$ 9 a/ronomy1 & are !re=e ( em95t & imm$iate practical utility3 o!rs !re are by :i* tables are produc$1 foll[+ ! mo/ extraord9>y1 & app>5tly caprici\s1 b /ill regul> laws4 ,?us a table w be comput$1 :i*1 to any requir$ ext5t1 %all co9cide ) a giv5 table1 & :i* %all deviate f t table = a s+le t]m1 or = any requir$ numb] ( t]ms1 & !n resume its c\rse1 or :i* %all p]man5tly alt] ! law ( its con/ruction4 ,?us ! eng9e has calculat$ a table :i* agre$ precisely ) a table ( squ>e numb]s1 until x atta9$ ! hundr$ & fir/ t]m1 :i* 0 n ! squ>e ( #aja1 nor 7 any ( ! subsequ5t numb]s squ>es4 ,aga91 x has comput$ a table :i* co9cid$ ) ! s]ies ( natural numb]s1 z f> z #ajj1jjj1jja1 b :i* subsequ5tly foll[$ ano!r law4 ,?is result 0 obta9$1 n by work+ ! eng9e ?r\gh ! :ole ( ! fir/ table1 = t w\ld h requir$ an enorm\s l5g? ( "t2 b by %[+1 f ! arrangem5t ( ! me*anism1 t x mu/ cont9ue to exhibit ! succession ( natural numb]s1 until x w\ld rea* #ajj1jjj1jjj4 ,to save "t1 ! eng9e 0 set by ! h& to ! numb] #iiiiiiie1 & 0 !n put 9 regul> op]ation4 ,x produc$ successively ! foll[+ numb]s4#ad #ii1iii1iif #ii1iii1iig #ii1iii1iih #ii1iii1iii #ajj1jjj1jjj #ajj1jaj1jjb #ajj1jcj1jjc #ajj1jfj1jjd #ajj1ajj1jje #ajj1aej1jjf c4 c4 ,footnote #ad3 ,su* results z ?is sugge/ a tra9 ( reflection on ! nature & op]ation ( g5]al laws1 :i* w\ld lead to v curi\s & int]e/+ speculations4 ,! natural philosoph] & a/ronom] w be h>dly less /ruck ) !m ?an ! metaphysician & !ologian4 ,equations h be5 already tabulat$ by ! portion ( ! ma*9]y :i* has be5 put toge!r1 :i* are s f> beyond ! rea* ( ! pres5t p[] ( ma!matics1 t no di/ant t]m ( ! table c be pr$ict$1 nor any function discov]$ capable ( express+ its g5]al law4 ,yet ! v fact ( ! table be+ produc$ by me*anism ( an inv>iable =m1 & includ+ a di/9ct pr9ciple ( me*anical action1 r5d]s x q manife/ t "s g5]al law mu/ exi/ 9 e table :i* x produces4 ,b we mu/ dismiss ^! speculations3 we feel x impossible to /ret* ! p[]s ( our own m9d1 s z to grasp ! probable capabilities ( ?is spl5did production ( comb9$ me*anical & ma!matical g5ius2 mu* less c we hope to enable o!rs to appreciate !m1 )\t be+ furni%$ ) su* means ( compreh5d+ !m z ^? ) :i* we h be5 fav\r$4 ,ye>s mu/ 9 fact elapse1 & _m enquir]s direct _! en]gies to ! cultivation ( ! va/ field ( rese>* ?us op5$1 be=e we c fully e/imate ! ext5t ( ?is triumph ( matt] ov] m9d4 ',nor is x1' says ,mr ,colebrooke1 'among ! lea/ curi\s results ( ?is +5i\s device1 t x af=ds a new op5+ = discov]y1 s9ce x is applicable1 z has be5 %[n by its inv5tor1 to surm\nt novel difficulties ( analysis4 ,n conf9$ to con/ant diff]5ces1 x is available 9 e case ( diff]5ces t foll[ a def9ite law1 r$ucible !re=e to an equation4 ,an eng9e adju/$ to ! purpose be+ set to "w1 w produce any di/ant t]m1 or succession ( t]ms1 requir$--?us pres5t+ ! num]ical solution ( a problem1 ev5 ?\gh ! analytical solution be yet undet]m9$4' ,t ! future pa? ( "s important bran*es ( ma!matical enquiry mu/ n[ 9 "s measure be direct$ by ! dictates ( me*anism1 is suffici5tly evid5t2 = :o w\ld toil on 9 any c\rse ( analytical enquiry1 9 :i* he mu/ ultimately dep5d on ! exp5sive & fallible aid ( human ari?metic1 ) an in/rum5t 9 8 h&s1 9 :i* all ! dull monotony ( num]ical computation is turn$ ov] to ! untir+ action & un]r+ c]ta9ty ( me*anical ag5cy8 ,x is wor? notice1 t ea* ( ! axes 9 front ( ! ma*9]y on :i* ! figure :eels revolve1 is connect$ ) a bell1 ! tongue ( :i* is gov]n$ by a sy/em ( lev]s1 mov$ by ! sev]al figure :eels2 an adju/m5t is provid$ by :i* ! lev]s %all be dismiss$1 s z to all[ ! hamm] to /rike aga9/ ! bell1 :5ev] any propos$ numb] %all be exhibit$ on ! axis4 ,?is contrivance enables ! ma*9e to give notice to its att5dants at any "t t an adju/m5t may be requir$4 ,among a great v>iety ( curi\s accid5tal prop]ties 7s to speak7 :i* ! ma*9e is f\nd to possess1 is "o by :i* x is capable ( solv+ num]ical equations :i* h rational roots4 ,su* an equation be+ r$uc$ 7z x always may be7 by suitable trans=mations to t /ate 9 :i* ! roots %all be :ole numb]s1 ! values #j1 #a1 #b1 #c1 c41 are sub/itut$ = ! unkn[n quantity1 & ! correspond+ values ( ! equation asc]ta9$4 ,f ^! a suffici5t numb] ( diff]5ces be+ d]iv$1 !y are set ^u ! ma*9e4 ,! ma*9e be+ !n put 9 motion1 ! table axis w exhibit ! successive values ( ! =mula1 correspond+ to ! sub/itutions ( ! successive :ole numb]s = ! unkn[n quantity3 at l5g? ! numb] exhibit$ on ! table axis w be #j1 :i* w evid5tly correspond to a root ( ! equation4 ,by previ\s adju/m5t1 ! bell ( ! table axis w 9 ?is case r+ & give notice ( ! exhibition ( ! value ( ! root 9 ano!r "p ( ! ma*9]y4 ,if ! equation h imag9>y roots1 ! =mula be+ necess>ily a maximum or m9imum on ! occurr5ce ( su* roots1 ! fir/ diff]5ce w become no?+2 & ! dials ( t axis w "u su* circum/ances pres5t to ! respective indices4 ,by previ\s adju/m5t1 ! bell ( ?is axis w\ld "h give notice ( a pair ( imag9>y roots4 ,mr ,colebrooke speculates on ! probable ext5sion ( ^! p[]s ( ! ma*9e3 ',x may n !re=e be deem$ too sangu9e an anticipation :5 ,i express ! hope t an in/rum5t :i*1 9 its simpl] =m1 atta9s to ! extraction ( roots ( numb]s1 & approximates to ! roots ( equations1 may1 9 a m advanc$ /ate ( improvem5t1 rise to ! approximate solution ( algebraic equations ( elevat$ degrees4 ,i ref] to solutions ( su* equations propos$ by ,la ,grange1 & m rec5tly by o!r annali/s1 :i* involve op]ations too t$i\s & intricate = use1 & :i* mu/ rema9 )\t efficacy1 unless "s mode be devis$ ( abridg+ ! lab\r1 or facilitat+ ! means ( its p]=mance4 ,9 any case ?is eng9e t5ds to light5 ! excessive & accumulat+ burd5 ( ari?metical application ( ma!matical =mul1 & to relieve ! progress ( sci5ce f :at is ju/ly t]m$ by ! au?or ( ?is inv5tion1 ! ov]:elm+ encumbrance ( num]ical detail4' ,al?\gh !re are n m ?an eighte5 figure :eels on ea* axis1 & !re=e x might be suppos$ t ! ma*9]y 0 capable ( calculat+ only to ! ext5t ( eighte5 decimal places2 yet !re are contrivances connect$ ) x1 by :i*1 9 two successive calculations1 x w be possible to calculate ev5 to ! ext5t ( ?irty decimal places4 ,its p[]s1 !re=e1 9 ?is respect1 greatly exce$ any :i* c be requir$ 9 practical sci5ce4 ,x is also rem>kable1 t ! ma*9]y is capable ( produc+ ! calculat$ results true to ! la/ figure4 ,we h already expla9$1 t :5 ! figure :i* w\ld foll[ ! la/ is great] ?an #d1 !n x w\ld be necess>y to increase ! la/ figure by #a2 s9ce ! excess ( ! calculat$ numb] above ! true value w\ld 9 su* case be less ?an its defect f x w\ld be1 _h ! regul>ly comput$ f9al figure be5 adopt$3 ?is is a precaution necess>y 9 all num]ical tables1 & x is "o :i* w\ld h>dly h be5 expect$ to be provid$ = 9 ! calculat+ ma*9]y4 ,z might be expect$ 9 a me*anical und]tak+ ( su* complexity & novelty1 _m practical difficulties h s9ce its comm5cem5t be5 enc\nt]$ & surm\nt$4 ,x might h be5 =ese51 t _m exp$i5ts w\ld be adopt$ & c>ri$ into effect1 :i* f>!r exp]im5ts w\ld r5d] x necess>y to reject2 & ?us a l>ge s\rce ( additional exp5se c\ld sc>cely fail to be produc$4 ,to a c]ta9 ext5t ?is has tak5 place2 b ow+ to ! admirable sy/em ( me*anical draw+s1 :i* 9 e in/ance ,mr ,babbage has caus$ to be made1 & ow+ to 8 own pr(\nd acqua9tance ) ! practical work+ ( ! mo/ complicat$ me*anism1 he has be5 able to pr$ict 9 e case :at ! result ( any contrivance w\ld be1 z p]fectly f ! draw+1 z if x _h be5 r$uc$ to ! =m ( a work+ model4 ,! draw+s1 consequ5tly1 =m a mo/ ext5sive & ess5tial "p ( ! ent]prise4 ,!y are execut$ ) extraord9>y ability & precision1 & may be consid]$ z p]haps ! be/ specim5s ( me*anical draw+s :i* h ev] be5 execut$4 ,x has be5 on ^!1 & on ^! only1 t ! "w ( inv5tion has be5 be/[$4 ,9 ^!1 all ^? progressive modifications sugge/$ by consid]ation & /udy h be5 made2 & x 0 n until ! inv5tor 0 fully satisfi$ ) ! result ( any contrivance1 t he _h x r$uc$ to a work+ =m4 ,! :ole ( ! loss :i* has be5 incurr$ by ! necess>ily progressive c\rse ( inv5tion1 has be5 ! exp5se ( reject$ draw+s4 ,no?+ c p]haps m =cibly illu/rate ! ext5t ( lab\r & ?\ght :i* has be5 incurr$ 9 ! production ( ?is ma*9]y1 ?an ! contemplation ( ! work+ draw+s :i* h be5 execut$ previ\sly to its con/ruction3 ^! draw+s cov] above a ?\s& squ>e feet ( surface1 & _m ( !m are ( ! mo/ elaborate & complicat$ description4 ,"o ( ! practical difficulties :i* pres5t$ !mselves at a v e>ly /age 9 ! progress ( ?is und]tak+1 0 ! impossibility ( be>+ 9 m9d all ! v>iety ( motions propagat$ simultane\sly ?r\gh s _m complicat$ tra9s ( me*anism4 ,no?+ b ! utmo/ imag9able h>mony & ord] among su* a numb] ( movem5ts1 c\ld prev5t ob/ructions aris+ f incompatible motions enc\nt]+ ea* o!r4 ,x 0 v soon f\nd impossible1 by a m]e act ( memory1 to gu>d aga9/ su* an occurr5ce2 & ,mr ,babbage f\nd1 t1 )\t "s effective exp$i5t by :i* he c\ld at a glance see :at e mov+ piece 9 ! ma*9]y 0 do+ at ea* in/ant ( "t1 su* inconsi/5cies & ob/ructions z are "h allud$ to mu/ cont9ually h occurr$4 ,?is difficulty 0 remov$ by ano!r inv5tion ( ev5 a m g5]al nature ?an ! calculat+ ma*9]y itself1 & pregnant ) results probably ( high] importance4 ,?is inv5tion consi/$ 9 ! contrivance ( a s*eme ( me*anical notation :i* is g5]ally applicable to all ma*9]y :atsoev]2 & :i* is exhibit$ on a table or plan consi/+ ( two di/9ct sections4 ,9 ! fir/ is trac$1 by a peculi> sy/em ( signs1 ! orig9 ( e motion :i* takes place ?r\gh\t ! ma*9]y2 s t ! me*ani/ or inv5tor is able1 by mov+ 8 f+] along a c]ta9 l9e1 to foll[ out ! motion ( e piece f effect to cause1 until he arrives at ! prime mov]4 ,! same sign :i* ?us indicates ! s\rce ( motion indicates likewise ! species ( motion1 :e!r x be cont9u\s or reciprocat+1 circul> or progressive1 c4 ,! same sy/em ( signs fur!r indicates ! nature ( ! me*anical connexion betwe5 ! mov] & ! ?+ mov$1 :e!r x be p]man5t & inv>iable 7z betwe5 ! two arms ( a lev]71 or :e!r ! mov] & ! mov$ are sep>ate & indep5d5t pieces1 z is ! case :5 a p9ion drives a :eel2 also :e!r ! motion ( "o piece necess>ily implies ! motion ( ano!r2 or :5 su* motion 9 ! "o is int]rupt$1 & 9 ! o!r cont9u\s1 c4 ,! second section ( ! table divides ! "t ( a complete p]iod ( ! ma*9]y into any requir$ numb] ( p>ts2 & x exhibits 9 a map1 z x 71 t :i* e "p ( ! ma*9e is do+ at ea* mom5t ( "t4 ,9 ?is way1 incompatibility 9 ! motions ( diff]5t p>ts is r5d]$ p]ceptible at a glance4 ,by su* means ! contriv] ( ma*9]y is n m]ely prev5t$ f introduc+ into "o "p ( ! me*anism any movem5t inconsi/5t ) ! simultane\s action ( ! o!r p>ts2 b :5 he f9ds t ! introduction ( any p>ticul> movem5t is necess>y = 8 purpose1 he c easily & rapidly exam9e ! :ole range ( ! ma*9]y dur+ "o ( its p]iods1 & c f9d by inspection :e!r !re is any1 & :at portion ( "t1 at :i* no motion exi/s incompatible ) ! desir$ "o1 & ?us discov] a ni*e1 z x 71 9 :i* to place ! requir$ movem5t4 ,a fur!r & collat]al advantage consi/s 9 plac+ x 9 ! p[] ( ! contriv] to ex]cise ! utmo/ possible economy ( "t 9 ! application ( 8 mov+ p[]4 ,= example1 )\t "s in/rum5t ( me*anical enquiry equally p[]ful ) t n[ describ$1 x w\ld be sc>cely possible1 at lea/ 9 ! fir/ in/ance1 s to arrange ! v>i\s movem5ts t !y %\ld be all execut$ 9 ! lea/ possible numb] ( revolutions ( ! mov+ axis4 ,additional revolutions w\ld almo/ inevitably be made = ! purpose ( produc+ movem5ts & *anges :i* x w\ld be possible to introduce 9 "s ( ! phases ( previ\s revolutions3 & !re is no "o acqua9t$ ) ! hi/ory ( me*anical inv5tion :o mu/ n be aw>e1 t 9 ! progressive contrivance ( almo/ e ma*9e ! e>lie/ arrangem5ts are inv>iably defective 9 ?is respect2 & t x is only by a succession ( improvem5ts1 sugge/$ by long exp]i5ce1 t t arrangem5t is at l5g? arriv$ at1 :i* accompli%es all ! necess>y motions 9 ! %orte/ possible "t4 ,by ! application ( ! me*anical notation1 h[ev]1 absolute p]fection may be arriv$ at 9 ?is respect2 ev5 be=e a s+le "p ( ! ma*9]y is con/ruct$1 & be=e x has any o!r exi/5ce ?an t :i* x obta9s ^u pap]4 ,examples ( ?is class ( advantages d]ivable f ! notation w occur to ! m9d ( e "o acqua9t$ ) ! hi/ory ( me*anical inv5tion4 ,9 ! common suction-pump1 = example1 ! effective ag5cy ( ! p[] is susp5d$ dur+ ! desc5t ( ! pi/on4 ,a v simple contrivance1 h[ev]1 w transf] to ! desc5t ! "w to be accompli%$ 9 ! next asc5t2 s t ! duty ( f\r /rokes ( ! pi/on may ?us be execut$ 9 ! "t ( two4 ,9 ! e>li] applications ( ! /eam-eng9e1 t ma*9e 0 appli$ almo/ exclusively to ! process ( pump+2 & ! p[] act$ only dur+ ! desc5t ( ! pi/on1 be+ susp5d$ dur+ its asc5t4 ,:51 h[ev]1 ! notion ( apply+ ! eng9e to ! g5]al purposes ( manufacture occurr$ to ! m9d ( ,watt1 he saw t x w\ld be necess>y to cause x to produce a cont9u$ rotatory motion2 &1 !re=e1 t ! int]vals ( int]mission mu/ be fill$ up by ! action ( ! p[]4 ,he fir/ propos$ to accompli% ?is by a second cyl9d] work+ alt]nately ) ! fir/2 b x soon became app>5t t ! blank :i* exi/$ dur+ ! up/roke 9 ! action ( ! p[]1 might be fill$ up by introduc+ ! /eam at bo? ends ( ! cyl9d] alt]nately4 ,_h ,watt plac$ be=e him a s*eme ( me*anical notation su* z we allude to1 ?is exp$i5t w\ld h be5 s obtrud$ ^u him t he mu/ h adopt$ x f ! fir/4 ,"o ( ! circum/ances f :i* ! me*anical notation d]ives a great portion ( its p[] z an in/rum5t ( inve/igation & discov]y1 is t x enables ! inv5tor to dismiss f 8 ?\ghts1 & to dis5cumb] 8 imag9ation ( ! arrangem5t & connexion ( ! me*anism2 :i*1 :5 x is v complex 7& x is 9 t case t ! notation is mo/ useful71 c only be kept be=e ! m9d by an emb>rass+ & pa9ful ef=t4 ,9 ?is respect ! p[]s ( ! notation may n inaptly be illu/rat$ by ! facilities d]iv$ 9 complex & difficult ari?metical que/ions f ! use ( ! language & notation ( algebra4 ,:5 once ! peculi> conditions ( ! "q are translat$ into algebraical signs1 & 'r$uc$ to an equation1' ! computi/ dismisses f 8 ?\ghts all ! circum/ances ( ! "q1 & is reliev$ f ! consid]ation ( ! complicat$ relations ( ! quantities ( v>i\s k9ds :i* may h ent]$ x4 ,he deals ) ! algebraical symbols1 :i* are ! repres5tatives ( ^? quantities & relations1 accord+ to c]ta9 te*nical rules ( a g5]al nature1 ! tru? ( :i* he has previ\sly e/abli%$2 &1 by a process almo/ me*anical1 he arrives at ! requir$ result4 ,:at algebra is to ari?metic1 ! notation we n[ allude to is to me*anism4 ,! v>i\s p>ts ( ! ma*9]y "u consid]ation be+ once express$ ^u pap] by prop] symbols1 ! enquir] dismisses altoge!r f 8 ?\ghts ! me*anism itself1 & att5ds only to ! symbols2 ! managem5t ( :i* is s extremely simple & obvi\s1 t ! mo/ unpractis$ p]son1 hav+ once acquir$ an acqua9tance ) ! signs1 _c fail to compreh5d _! use4 ,a rem>kable in/ance ( ! p[] & utility ( ?is notation occurr$ 9 a c]ta9 /age ( ! inv5tion ( ! calculat+ ma*9]y4 ,a "q arose z to ! be/ me?od ( produc+ & arrang+ a c]ta9 s]ies ( motions necess>y to pr9t & calculate a numb]4 ,! inv5tor1 assi/$ by a practical eng9e] ( consid]able exp]i5ce & skill1 _h s arrang$ ^! motions1 t ! :ole might be p]=m$ by twelve revolutions ( ! pr9cipal mov+ axis4 ,x seem$1 h[ev]1 desirable1 if possible1 to execute ^! motions by a less numb] ( revolutions4 ,to accompli% ?is1 ! eng9e] sat d[n to /udy ! complicat$ details ( a "p ( ! ma*9]y :i* _h be5 put toge!r2 ! inv5tor at ! same "t appli$ himself to ! consid]ation ( ! arrangem5t & connexion ( ! symbols 9 8 s*eme ( notation4 ,aft] a %ort "t1 by "s transposition ( symbols1 he caus$ ! receiv$ motions to be complet$ by eight turns ( ! axis4 ,?is he accompli%$ by transf]r+ ! symbols :i* occupi$ ! la/ f\r divisions ( 8 s*eme1 into su* blank spaces z he c\ld discov] 9 ! fir/ eight divisions2 due c>e be+ tak5 t no symbols %\ld express actions at once simultane\s & incompatible4 ,pu%+ 8 enquiry1 h[ev]1 /ill fur!r1 he proce$$ to asc]ta9 :e!r 8 s*eme ( symbols did n admit ( a /ill m compact arrangem5t1 & :e!r eight revolutions 7 n m ?an en\gh to accompli% :at 0 requir$4 ,"h ! p[]s ( ! practical eng9e] completely broke d[n4 ,by no ef=t c\ld he br+ be=e 8 m9d su* a view ( ! complicat$ me*anism z w\ld enable him to decide ^u any improv$ arrangem5t4 ,! inv5tor1 h[ev]1 )\t any extraord9>y m5tal ex]tion1 & m]ely by slid+ a bit ( rul$ pa/ebo>d up & d[n 8 plan1 9 se>* ( a vacancy :]e ! diff]5t motions might be plac$1 at l5g? contriv$ to pack all ! motions1 :i* _h previ\sly occupi$ eight turns ( ! h&le1 into five turns4 ,! symbolic in/rum5t ) :i* he conduct$ ! inve/igation1 n[ in=m$ him ( ! impossibility ( r$uc+ ! action ( ! ma*9e to a m cond5s$ =m4 ,?is appe>$ by ! fulness ( e space along ! l9es ( compatible action4 ,x 01 h[ev]1 /ill possible1 by go+ back to ! actual ma*9]y1 to asc]ta9 :e!r movem5ts1 :i*1 "u exi/+ arrangem5ts1 7 incompatible1 might n be br\ght into h>mony4 ,?is he accord+ly did1 & succe$$ 9 dim9i%+ ! numb] ( incompatible conditions1 & !reby r5d]$ x possible to make actions simultane\s :i* 7 be=e necess>ily successive4 ,! notation 0 n[ aga9 call$ into requisition1 & a new disposition ( ! p>ts 0 made4 ,at ?is po9t ( ! inve/igation1 ?is extraord9>y in/rum5t ( me*anical analysis put =? "o ( its mo/ s+ul> ex]tions ( p[]4 ,x pres5t$ to ! eye ( ! eng9e] two curr5ts ( me*anical action1 :i*1 f _! nature1 c\ld n be simultane\s2 & ea* ( :i* occupi$ a complete revolution ( ! axis1 except ab\t a tw5tie?2 ! "o occupy+ ! la/ n9ete5-tw5tie?s ( a complete revolution ( ! axis1 & ! o!r occupy+ ! fir/ n9ete5-tw5tie?s ( a complete revolution4 ,"o ( ^! /reams ( action 01 ! successive pick+ up by ! c>ry+ f+]s ( ! successive c>ry+ claws2 & ! o!r 01 ! successive %oot+ ( n9ete5 bolts by ! n9ete5 bolt+ f+]s4 ,! notation r5d]$ x obvi\s1 t z ! bolt+ action comm5c$ a small space bel[ ! comm5cem5t ( ! c>ry+1 & end$ an equal space bel[ ! t]m9ation ( ! c>ry+1 ! two /reams ( action c\ld be made to fl[ aft] "o ano!r 9 "o & ! same revolution ( ! axis4 ,he ?us succe$$ 9 r$uc+ ! p]iod ( complet+ ! action to f\r turns ( ! axis2 :5 ! notation aga9 in=m$ him t he _h aga9 atta9$ a limit ( cond5s$ action1 :i* c\ld n be exce$$ )\t a fur!r *ange 9 ! me*anism4 ,to ! me*anism he aga9 recurr$1 & soon f\nd t x 0 possible to introduce a *ange :i* w\ld cause ! action to be complet$ 9 ?ree revolutions ( ! axis4 ,an odd numb] ( revolutions1 h[ev]1 be+ att5d$ ) c]ta9 practical inconv5i5ces1 x 0 consid]$ m advantage\s to execute ! motions 9 f\r turns2 & "h aga9 ! notation put =? its p[]s1 by in=m+ ! inv5tor1 ?r\gh ! eye1 almo/ indep5d5t ( 8 m9d1 :at w\ld be ! mo/ elegant1 symmetrical1 & h>moni\s disposition ( ! requir$ motions 9 f\r turns4 ,?is application ( an almo/ metaphysical sy/em ( ab/ract signs1 by :i* ! motion ( ! h& p]=ms ! (fice ( ! m9d1 & ( pr(\nd practical skill 9 me*anics alt]nately1 to ! con/ruction ( a mo/ complicat$ eng9e1 =cibly rem9ds u ( a p>allel 9 ano!r sci5ce1 :]e ! *emi/ ) difficulty succe$s 9 dissolv+ a refractory m9]al1 by ! alt]nate action ( ! mo/ p[]ful acids1 & ! mo/ cau/ic alkalies1 repeat$ 9 long-cont9u$ succession4 ,?is important discov]y 0 expla9$ by ,mr ,babbage1 9 a %ort pap] read be=e ! ,royal ,society1 & publi%$ 9 ! ,philosophical ,transactions 9 #ahbf4ae ,x is to u m a matt] ( regret ?an surprise1 t ! subject did n receive f sci5tific m5 9 ?is c\ntry t att5tion to :i* its importance 9 e practical po9t ( view s fully entitl$ x4 ,to appreciate x w\ld inde$ h be5 sc>cely possible1 f ! v brief memoir :i* its inv5tor pres5t$1 unaccompani$ by any obs]vations or argum5ts ( a nature to =ce x ^u ! att5tion ( m9ds unprep>$ = x by ! nature ( _! /udies or occupations4 ,9 ?is c\ntry1 sci5ce has be5 g5]ally sep>at$ f practical me*anics by a wide *asm4 ,x w be easily admitt$1 t an assembly ( em95t naturali/s & physicians1 ) a spr9kl+ ( a/ronom]s1 & "o or two ab/ract ma!maticians1 7 n precisely ! p]sons be/ qualifi$ to appreciate su* an in/rum5t ( me*anical inve/igation z we h "h describ$4 ,we %all n !re=e be und]/ood z int5d+ ! slighte/ disrespect = ^! di/+ui%$ p]sons1 :5 we express our regret1 t a discov]y ( su* p>am\nt practical value1 9 a c\ntry preem95tly conspicu\s = ! results ( its ma*9]y1 %\ld fall /ill-born & inconsequ5tial ?r\gh _! h&s1 & be buri$ unhon\r$ & undiscrim9at$ 9 _! miscellane\s transactions4 ,we tru/ t a m auspici\s p]iod is at h&2 t ! *asm :i* has sep>at$ practical f sci5tific m5 w spe$ily close2 & t t comb9ation ( k w be effect$1 :i* c only be obta9$ :5 we see ! m5 ( sci5ce m frequ5tly ext5d+ _! obs]vant eye ov] ! wond]s ( our factories1 & our great practical manufactur]s1 ) a reciprocal ambition1 pres5t+ !mselves z active & useful memb]s ( our sci5tific associations4 ,:5 ?is has tak5 place1 an ord] ( sci5tific m5 w spr+ up1 :i* w r5d] impossible an ov]sight s little cr$itable to ! c\ntry z t :i* has be5 committ$ respect+ ! me*anical notation4#af ,?is notation has rec5tly und]gone v consid]able ext5sion & improvem5t4 ,an additional section has be5 introduc$ into x2 design$ to express ! process ( circulation 9 ma*9es1 ?r\gh :i* fluids1 :e!r liquid or gase\s1 are mov$4 ,mr ,babbage1 ) ! assi/ance ( a fri5d :o happ5$ to be conv]sant ) ! /ructure & op]ation ( ! /eam-eng9e1 has illu/rat$ x ) s+ul> felicity & success 9 its application to t ma*9e4 ,an em95t ,fr5* surgeon1 on see+ ! s*eme ( notation ?us appli$1 imm$iately sugge/$ ! advantages :i* mu/ att5d x z an in/rum5t = express+ ! /ructure1 op]ation1 & circulation ( ! animal sy/em2 & we ent]ta9 no d\bt ( its adequacy = t purpose4 ,n only ! me*anical connexion ( ! solid memb]s ( ! bodies ( m5 & animals1 b likewise ! /ructure & op]ation ( ! s(t] p>ts1 includ+ ! muscles1 integum5ts1 membranes1 c42 ! nature1 motion1 & circulation ( ! v>i\s fluids1 _! reciprocal effects1 ! *anges ?r\gh :i* !y pass1 ! deposits :i* !y leave 9 v>i\s p>ts ( ! sy/em2 ! functions ( respiration1 dige/ion1 & assimilation1--all w\ld f9d appropriate symbols & repres5tatives 9 ! notation1 ev5 z x n[ /&s1 )\t ^? additions ( :i*1 h[ev]1 x is easily susceptible4 ,inde$1 :5 we reflect = :at a v diff]5t purpose ?is s*eme ( symbols 0 contriv$1 we _c refra9 f express+ our wond] t x %\ld seem1 9 all respects1 z if x _h be5 design$ expressly = ! purposes ( anatomy & physiology4 ,footnote #ae3 ,phil4 ,trans4 #ahbj1 ,"p ,,iii4 p4 #bej1 on a me?od ( express+ by signs ! action ( ma*9]y4 ,footnote #af3 ,?is discov]y has be5 m ju/ly appreciat$ by sci5tific m5 abroad4 ,x 01 almo/ imm$iately aft] its publication1 adopt$ z ! topic ( lectures1 9 an in/itution on ! ,cont95t = ! in/ruction ( ,civil ,eng9e]s4 ,ano!r ( ! uses :i* ! slighte/ att5tion to ! details ( ?is notation irresi/ibly =ces ^u our notice1 is to exhibit1 9 ! =m ( a connect$ plan or map1 ! organization ( an ext5sive factory1 or any great public in/itution1 9 :i* a va/ numb] ( individuals are employ$1 & _! duties regulat$ 7z !y g5]ally are or ought to be7 by a consi/5t & well-dige/$ sy/em4 ,! me*anical notation is admirably adapt$1 n only to express su* an organiz$ connexion ( human ag5ts1 b ev5 to sugge/ ! improvem5ts ( :i* su* organization is susceptible--to betray its weak & defective po9ts1 & to disclose1 at a glance1 ! orig9 ( any fault :i* may1 f "t to "t1 be obs]v$ 9 ! work+ ( ! sy/em4 ,our limits1 h[ev]1 preclude u f pursu+ ?is int]e/+ topic to ! ext5t :i* its importance w\ld ju/ify4 ,we %all be satisfi$ if ! h9ts "h ?r[n out %\ld direct to ! subject ! att5tion ( ^? :o1 be+ mo/ int]e/$ 9 su* an enquiry1 are likely to prosecute x ) greate/ success4 ,"o ( ! consequ5ces :i* has aris5 9 ! prosecution ( ! inv5tion ( ! calculat+ ma*9]y1 has be5 ! discov]y ( a multitude ( me*anical contrivances1 :i* h be5 elicit$ by ! exig5cies ( ! und]tak+1 & :i* are z novel 9 _! nature z ! purposes 7 novel :i* !y 7 design$ to atta94 ,9 "s cases sev]al diff]5t contrivances 7 devis$ = ! atta9m5t ( ! same end2 & t among !m :i* 0 be/ suit$ = ! purpose 0 f9ally select$3 ! reject$ exp$i5ts--^? ov]fl[+s or wa/e ( ! inv5tion--7 n1 h[ev]1 always f\nd useless4 ,l ! wa/e 9 v>i\s manufactures1 !y 7 soon conv]t$ to purposes ( utility4 ,^! reject$ contrivances h f\nd _! way1 9 _m cases1 into ! mills ( our manufactur]s2 & we n[ f9d !m busily effect+ purposes1 f> diff]5t f any :i* ! inv5tor dream$ (1 9 ! sp9n+-frames ( ,man*e/]4#ag ,footnote #ag3 ,an em95t & weal?y retir$ manufactur] at ,man*e/] assur$ u1 t on ! occasion ( a visit to ,london1 :5 he 0 fav\r$ ) a view ( ! calculat+ ma*9]y1 he f\nd 9 x me*anical contrivances1 :i* he subsequ5tly introduc$ ) ! greate/ advantage into 8 own sp9n+-ma*9]y4 ,ano!r dep>tm5t ( me*anical art1 :i* has be5 enri*$ by ?is inv5tion1 has be5 t ( tools4 ,! great v>iety ( new =ms :i* x 0 necess>y to produce1 creat$ ! necessity ( contriv+ & con/ruct+ a va/ numb] ( novel & mo/ valuable tools1 by :i*1 ) ! aid ( ! la!1 & t alone1 ! requir$ =ms c\ld be giv5 to ! diff]5t p>ts ( ! ma*9]y ) all ! requisite accuracy4 ,! idea ( calculation by me*anism is n new4 ,ari?metical in/rum5ts1 su* z ! calculat+ bo>ds ( ! anci5ts1 on :i* !y made _! computations by ! aid ( c\nt]s--! ,abacus1 an in/rum5t = comput+ by ! aid ( balls slid+ ^u p>allel rods--! me?od ( calculation inv5t$ by ,b>on ,napi]1 call$ by him ,rhabdology1 & s9ce call$ ,napi]'s bones--! ,swan ,pan ( ! ,*9ese--& o!r simil> contrivances1 among :i* m p>ticul>ly may be m5tion$ ! ,slid+ ,rule1 ( s mu* use 9 practical calculations to mod]n eng9e]s1 w occur to e read]3 ^! may m prop]ly be call$ ari?metical in/rum5ts1 p>tak+ m or less ( a me*anical "*4 ,b ! e>lie/ piece ( me*anism to :i* ! "n ( a 'calculat+ ma*9e' c fairly be giv51 appe>s to h be5 a ma*9e inv5t$ by ! celebrat$ ,pascal4 ,?is philosoph] & ma!matician1 at a v e>ly age1 be+ engag$ ) 8 fa!r1 :o held an (ficial situation 9 ,upp] ,norm&y1 ! duties ( :i* requir$ frequ5t num]ical calculations1 contriv$ a piece ( me*anism to facilitate ! p]=mance ( !m4 ,?is me*anism consi/$ ( a s]ies ( :eels1 c>ry+ cyl9drical b>rels1 on :i* 7 engrav$ ! t5 ari?metical *>act]s1 9 a mann] n v dissimil> to t already describ$4 ,! :eel :i* express$ ea* ord] ( units 0 s connect$ ) ! :eel :i* express$ ! sup]ior ord]1 t :5 ! =m] pass$ f #i to #j1 ! latt] 0 necess>ily advanc$ "o figure2 & ?us ! process ( c>ry+ 0 execut$ by me*anism3 :5 "o numb] 0 to be add$ to ano!r by ?is ma*9e1 ! addition ( ea* figure to ! o!r 0 p]=m$ by ! h&2 :5 x 0 requir$ to add m ?an two numb]s1 ! additions 7 p]=m$ 9 ! same mann] successively2 ! second 0 add$ to ! fir/1 ! ?ird to _! sum1 & s on4 ,subtraction 0 r$uc$ to addition by ! me?od ( ari?metical complem5ts2 multiplication 0 p]=m$ by a succession ( additions2 & division by a succession ( subtractions4 ,9 all cases1 h[ev]1 ! op]ations 7 execut$ f :eel to :eel by ! h&4#ah ,footnote #ah3 ,see a description ( ?is ma*9e by ,did]ot1 9 ! ,encyc4 ,me?od42 also 9 ! works ( ,pascal1 tom4 ,,iv41 p4 #g2 ,p>is1 #ahai4 ,?is me*anism1 :i* 0 inv5t$ ab\t ! ye> #afej1 does n appe> ev] to h be5 br\ght into any practical use2 & seems to h spe$ily f\nd its appropriate place 9 a museum ( curiosities4 ,x 0 capable ( p]=m+ only p>ticul> ari?metical op]ations1 & ^! subject to all ! *ances ( error 9 manipulation2 att5d$ also ) little m exp$ition 7if s mu*71 z w\ld be atta9$ by ! p5 ( an exp]t comput]4 ,?is attempt ( ,pascal 0 foll[$ by v>i\s o!rs1 ) v little improvem5t1 & ) no additional success4 ,pol5us1 a le>n$ & +5i\s ,italian1 inv5t$ a ma*9e by :i* multiplication 0 p]=m$1 b :i* does n appe> to h af=d$ any mat]ial facilities1 nor any m security aga9/ error ?an ! common process ( ! p54 ,a simil> attempt 0 made by ,sir ,samuel ,morel&1 :o is describ$ z hav+ transf]r$ to :eel-"w ! figures ( ,napi]'s bones1 & z hav+ made "s additions to ! ma*9e ( ,pascal4#ai ,footnote #ai3 ,equidem ,morel&us 9 ,anglia1 tub /5tori au?or1 ,rhabdologiam ex baculis 9 cyl9drulos tran/ulit1 et additiones auxili>es p]agit 9 adjuncta ma*9a additionum ,pascaliana4 ,grillet1 a ,fr5* me*anician1 made a l attempt ) z little success4 ,ano!r contrivance = me*anical calculation 0 made by ,saund]son4 ,me*anical contrivances = p]=m+ p>ticul> ari?metical processes 7 also made ab\t a c5tury ago by ,deleprne & ,boitiss5deau2 b !y 7 m]ely modifications ( ,pascal's1 )\t v>y+ or ext5d+ its objects4 ,b "o ( ! mo/ rem>kable attempts ( ?is k9d :i* has be5 made s9ce t ( ,pascal1 0 a ma*9e inv5t$ by ,leibnitz1 ( :i* we are n aw>e t any detail$ or intelligible description 0 ev] publi%$4 ,leibnitz describ$ its mode ( op]ation1 & its results1 9 ! ,b]l9 ,miscellany1#bj b he appe>s to h decl9$ any description ( its details4 ,9 a lett] address$ by him to ,b]n\lli1 9 answ] to a reque/ ( ! latt] t he w\ld af=d a description ( ! ma*9]y1 he says1 ',descriptionem ejus d>e accuratam res non facilis =et4 ,de effectu ex eo judicav]is quod ad multiplic&um num]um sex figur>um1 e4g4 rotam quamdam tantum sexies gyr>i necesse e/1 nulla alia op]a m5tis1 nullis additionibus int]v5i5tibus2 quo facto1 integrum absolutumque productum oculis objicietur4'#ba ,he goes on to say t ! process ( division is p]=m$ indep5d5tly ( a succession ( subtractions1 su* z t us$ by ,pascal4 ,footnote #bj3 ,tom4 ,i41 p4 #cag4 ,footnote #ba3 ,com4 ,epi/4 tom4 ,i41 p4 #bhi4 ,x appe>s t ?is ma*9e 0 "o ( an extremely complicat$ nature1 :i* w\ld be att5d$ ) consid]able exp5se ( con/ruction1 & only fit to be us$ 9 cases :]e num]\s & exp5sive calculations 7 necess>y4#bb ,leibnitz obs]ves to 8 correspond5t1 :o requir$ :e!r x might n be br\ght into common use1 ',non e/ facta pro 8 qui ol]a aut pisculos v5dunt1 s$ pro obs]vatoriis aut cam]is computorum1 aut aliis1 qui sumptus facile f]unt et multo calculo eg5t4' ,nev]!less1 x does n appe> t ?is contrivance1 ( :i* ! inv5tor /ates t he caus$ two models to be made1 0 ev] appli$ to any useful purpose2 nor inde$ d ! me*anical details ( ! inv5tion appe> ev] to h be5 publi%$4 ,footnote #bb3 ,s$ ma*9am esse sumptuosam et mult>um rot>um in/> horologii3 ,huyg5ius aliquoties admonuit ut absolvi cur>em2 quod non s9e magno sumptu tdioque factum e/1 dum v>ie mihi cum opificibus fuit conflict&um4--,com4 ,epi/4 ,ev5 _h ! me*anism ( ^! ma*9es p]=m$ all :i* _! inv5tors expect$ f !m1 !y w\ld h be5 /ill altoge!r inapplicable = ! purposes to :i* x is propos$ t ! calculat+ ma*9]y ( ,mr ,babbage %all be appli$4 ,!y 7 all con/ruct$ ) a view to p]=m p>ticul> ari?metical op]ations1 & 9 all ( !m ! accuracy ( ! result dep5d$ m or less ^u manipulation4 ,! pr9ciple ( ! calculat+ ma*9]y ( ,mr ,babbage is p]fectly g5]al 9 its nature1 n dep5d+ on any p>ticul> ari?metical op]ation1 & is equally applicable to num]ical tables ( e k9d4 ,?is di/+ui%+ *>act]i/ic 0 well express$ by ,mr ,colebrooke 9 8 address to ! ,a/ronomical ,society on ?is inv5tion4 ',! pr9ciple :i* ess5tially di/+ui%es ,mr ,babbage's inv5tion f all ^! is1 t x proposes to calculate a s]ies ( numb]s foll[+ any law1 by ! aid ( diff]5ces1 & t by sett+ a few figures at ! outset2 a long s]ies ( numb]s is readily produc$ by a me*anical op]ation4 ,! me?od ( diff]5ces 9 a v wide s5se is ! ma!matical pr9ciple ( ! contrivance4 ,a ma*9e to add a numb] ( arbitr>y figures toge!r is no economy ( "t or tr\ble1 s9ce ea* individual figure mu/ be plac$ 9 ! ma*9e2 b x is o!rwise :5 ^? figures foll[ "s law4 ,! ins]tion ( a few at fir/ det]m9es ! magnitude ( ! next1 & ^? ( ! succe$+4 ,x is ?is con/ant repetition ( simil> op]ations :i* r5d]s ! computation ( tables a fit subject = ! application ( ma*9]y4 ,mr ,babbage's inv5tion puts an eng9e 9 ! place ( ! comput]2 ! "q is set to ! in/rum5t1 or ! in/rum5t is set to ! "q1 & by simply giv+ x motion ! solution is wr\ght1 & a /r+ ( answ]s is exhibit$4' ,b p]haps ! greate/ ( its advantages is1 t x pr9ts :at x calculates2 & ?is completely precludes ! possibility ( error 9 ^? num]ical results :i* pass into ! h&s ( ! public4 ',! usefulness ( ! in/rum5t1' says ,mr ,colebrooke1 'is ?us m ?an d\bl$2 = x n only saves "t & tr\ble 9 transcrib+ results into a tabul> =m1 & sett+ types = ! pr9t+ ( ! table1 b x likewise accompli%es ! yet m important object ( ensur+ accuracy1 obviat+ num]\s s\rces ( error ?r\gh ! c>eless h&s ( transcrib]s & compositors4' ,"s solicitude w d\btless be felt respect+ ! pres5t /ate ( ! calculat+ ma*9]y1 & ! probable p]iod ( its completion4 ,9 ! beg9n+ ( ! ye> #ahbi1 ,gov]nm5t direct$ ! ,royal ,society to in/itute su* enquiries z w\ld enable !m to report ^u ! /ate to :i* x _h !n arriv$2 & also :e!r ! progress made 9 its con/ruction confirm$ !m 9 ! op9ion :i* !y _h =m]ly express$1--t x w\ld ultimately prove adequate to ! important object :i* x 0 int5d$ to atta94 ,! ,royal ,society1 9 accordance ) ^! directions1 appo9t$ a ,committee to make ! necess>y enquiry1 & report4 ,?is ,committee consi/$ ( ,mr ,davies ,gilb]t1 !n ,presid5t1 ! ,secret>ies1 ,sir ,john ,h]s*el1 ,mr ,francis ,baily1 ,mr ,brunel1 eng9e]1 ,mr ,donk91 eng9e]1 ,mr ,g4 ,r5nie1 eng9e]1 ,mr ,b>ton1 comptroll] ( ! ,m9t1 & ,mr ,w>burton1 ,m4,p4 ,! volum9\s draw+s1 ! v>i\s tools1 & ! portion ( ! ma*9]y !n execut$1 und]w5t a close & elaborate exam9ation by ?is ,committee1 :o report$ ^u x to ! ,society4 ,!y /at$ 9 _! report1 t !y decl9$ ! consid]ation ( ! pr9ciple on :i* ! practicability ( ! ma*9]y dep5ds1 & ( ! public utility ( ! object :i* x proposes to atta92 because !y consid]$ ! =m] fully admitt$1 & ! latt] obvi\s to all :o consid] ! imm5se advantage ( accurate num]ical tables 9 all matt]s ( calculation1 especially 9 ^? :i* relate to a/ronomy & navigation1 & ! great v>iety & ext5t ( ^? :i* x is pr(ess$ly ! object ( ! ma*9]y to calculate & pr9t ) p]fect accuracy2--t absolute accuracy be+ "o ( ! prom95t pret5sions ( ! und]tak+1 !y _h direct$ _! att5tion especially to ?is po9t1 by c>eful exam9ation ( ! draw+s & ( ! "w already execut$1 & by repeat$ conf]5ces ) ,mr ,babbage on ! subject2--t ! result ( _! enquiry 01 t su* precautions appe>$ to h be5 tak5 9 e "p ( ! contrivance1 & s fully aw>e 0 ! inv5tor ( e circum/ance :i* might by possibility produce error1 t !y _h no hesitation 9 /at+ _! belief t ^! precautions 7 effectual1 & t :atev] ! ma*9e w\ld d1 x w\ld d truly4 ,!y fur!r /at$1 t ! progress :i* ,mr ,babbage _h !n made1 consid]+ ! v great difficulties to be ov]come 9 an und]tak+ ( s novel a k9d1 fully equall$ any expectations t c\ld reasonably h be5 =m$2 & t al?\gh sev]al ye>s _h elaps$ s9ce ! comm5cem5t ( ! und]tak+1 yet :5 ! necessity ( con/ruct+ plans1 sections1 elevations1 & work+ draw+s ( e "p2 ( con/ruct+1 & 9 _m cases inv5t+1 tools & ma*9]y ( great exp5se & complexity1 necess>y to =m ) ! requisite precision p>ts ( ! app>atus diff]+ f any :i* _h previ\sly be5 introduc$ 9 ord9>y me*anical works2 ( mak+ _m trials to asc]ta9 ! value ( ea* propos$ contrivance2 ( alt]+1 improv+1 & simplify+ ! draw+s2--t1 consid]+ all ^! matt]s1 ! ,committee1 in/ead ( feel+ surprise at ! "t :i* ! "w has occupi$1 felt m dispos$ to wond] at ! possibility ( accompli%+ s mu*4 ,! ,committee express$ _! confid5t op9ion ( ! adequacy ( ! ma*9]y to "w "u all ! friction & /ra9 to :i* x c be expos$2 ( its durability1 /r5g?1 solidity1 & equilibrium2 ( ! prev5tion (1 or comp5sation =1 we> by friction2 ( ! accuracy ( ! v>i\s adju/m5ts2 & ( ! judgm5t & discretion display$ by ! inv5tor1 9 8 det]m9ation to admit into ! me*anism no?+ b ! v be/ & mo/ f9i%$ workman%ip2 z a contr>y c\rse w\ld h be5 false economy1 & might h l$ to ! loss ( ! :ole capital exp5d$ on x4 ,f9ally1 consid]+ all t _h come be=e !m1 & rely+ on ! tal5t & skill display$ by ,mr ,babbage z a me*ani/ 9 ! progress ( ?is ardu\s und]tak+1 n less = :at rema9$1 ?an on ! matur$ & dige/$ plan & admirable execution ( :at is complet$1 ! ,committee did n hesitate to express _! op9ion1 t 9 ! !n /ate ( ! eng9e1 !y reg>d$ x z likely to fulfil ! expectations ent]ta9$ ( x by its inv5tor4 ,?is report 0 pr9t$ 9 ! comm5cem5t ( ! ye> #ahbi4 ,f t "t until ! beg9n+ ( ! ye> #ahcc1 ! progress ( ! "w has be5 sl[ & int]rupt$4 ,mean:ile _m unf\nd$ rum\rs h obta9$ circulation z to ! c\rse adopt$ by ,gov]nm5t 9 ?is und]tak+2 & z to ! position 9 :i* ,mr ,babbage /&s ) respect to x4 ,we %all "h /ate1 ^u au?ority on :i* ! mo/ p]fect reliance may be plac$1 :at h be5 ! actual circum/ances ( ! arrangem5t :i* has be5 made1 & ( ! /eps :i* h be5 already tak54 ,be+ advis$ t ! objects ( ! project$ ma*9]y 7 ( p>am\nt national importance to a m>itime c\ntry1 & t1 f its nature1 x c\ld nev] be und]tak5 ) advantage by any individual z a pecuni>y speculation1 ,gov]nm5t det]m9$ to engage ,mr ,babbage to con/ruct ! calculat+ eng9e = ! nation4 ,x 0 !n ?\ght t ! "w c\ld be complet$ 9 two or ?ree ye>s2 & x 0 accord+ly und]tak5 on ?is und]/&+ ab\t ! ye> #ahba1 & s9ce !n has be5 9 progress4 ,! execution ( ! workman%ip 0 confid$ to an eng9e] by :om all ! subord9ate workm5 7 employ$1 & :o suppli$ = ! "w ! requisite tools & o!r ma*9]y2 ! latt] be+ 8 own prop]ty1 & n t ( ,gov]nm5t4 ,?is eng9e] furni%$1 at int]vals1 8 acc\nts1 :i* 7 duly audit$ by prop] p]sons appo9t$ = t purpose4 ,x 0 ?\ght advisable--) a view1 p]haps1 to inve/ ,mr ,babbage ) a m /rict au?ority ov] ! subord9ate ag5ts--t ! paym5ts ( ^! acc\nts ( ! eng9e] %\ld pass ?r\gh 8 h&s4 ,! am\nt 0 accord+ly f "t to "t issu$ to him by ! ,treasury1 & paid ov] to ! eng9e]4 ,?is circum/ance has giv5 rise to reports1 t he has receiv$ consid]able sums ( money z a remun]ation = 8 skill & lab\r 9 inv5t+ & con/ruct+ ?is ma*9]y4 ,su* reports are altoge!r de/itute ( tru?4 ,he has receiv$1 nei!r directly nor indirectly1 any remun]ation :atev]2--on ! contr>y1 ow+ to v>i\s (ficial delays 9 ! issues ( money f ! ,treasury = ! paym5t ( ! eng9e]1 he has frequ5tly be5 oblig$ to advance ^! paym5ts himself1 t ! "w might proce$ )\t int]ruption4 ,_h he n be5 enabl$ to d ?is f 8 private res\rces1 x w\ld h be5 impossible t ! ma*9]y c\ld h arriv$ at its pres5t advanc$ /ate4 ,x w be a matt] ( regret to e fri5d ( sci5ce to le>n1 t1 not)/&+ su* assi/ance1 ! progress ( ! "w has be5 susp5d$1 & ! workm5 dismiss$ = m ?an a ye> & a half2 nor does !re at ! pres5t mom5t appe> to be any imm$iate prospect ( its be+ resum$4 ,:at ! causes may be ( a susp5sion s extraord9>y1 ( a project ( su* great national & univ]sal int]e/1--9 :i* ! c\ntry has already inve/$ a sum ( su* s]i\s am\nt z ,l4#ae1jjj1--;is a "q :i* w at once sugge/ itself to e m9d2 & is "o to :i*1 not)/&+ frequ5t enquiries1 9 qu>t]s f :i* correct in=mation might be expect$1 we h n be5 able to obta9 any satisfactory answ]4 ,x is n true1 we are assur$1 t ! ,gov]nm5t object to make ! necess>y paym5ts1 or ev5 advances1 to c>ry on ! "w4 ,x is n true1 we also are assur$1 t any practical difficulty has aris5 9 ! con/ruction ( ! me*anism2--on ! contr>y1 ! draw+s ( all ! p>ts ( x are complet$1 & may be inspect$ by any p]son appo9t$ on ! "p ( ,gov]nm5t to exam9e !m4#bc ,mr ,babbage is kn[n z a man ( unwe>i$ activity1 & aspir+ ambition4 ,:y1 !n1 x may be ask$1 is x t he1 see+ 8 pres5t reputation & future fame dep5d+ 9 s great a degree ^u ! successful issue ( ?is und]tak+1 has nev]!less all[$ x to /& /ill = s long a p]iod1 )\t di/9ctly po9t+ out to ,gov]nm5t ! c\rse :i* !y %\ld adopt to remove ! causes ( delay8 ,_h he done ?is 7:i* we consid] to be equally due to ! nation & to himself71 he w\ld h ?r[n ^u ,gov]nm5t & its ag5ts ! :ole responsibility = ! delay & consequ5t loss2 b we believe he has n done s4 ,on ! contr>y1 x is said t he has ( late almo/ )drawn f all int]f]5ce on ! subject1 ei!r ) ! ,gov]nm5t or ! eng9e]4 ,does n ,mr ,babbage p]ceive ! inf]5ce :i* ! _w w draw f ?is c\rse ( conduct8 ,does he n see t !y w impute x to a di/ru/ ( 8 own p[]1 or ev5 to a consci\sness ( 8 own inability to complete :at he has begun8 ,we feel assur$ t su* is n ! case2 & we are anxi\s1 equally = ! sake ( sci5ce1 & = ,mr ,babbage's own reputation1 t ! my/]y--= su* x mu/ be reg>d$--%\ld be cle>$ up2 & t all ob/ructions to ! progress ( ! und]tak+ %\ld imm$iately be remov$4 ,does ?is sup95ess & app>5t indiff]5ce1 s incompatible ) ! kn[n "* ( ,mr ,babbage1 arise f any feel+ ( dissatisfaction at ! exi/+ arrangem5ts betwe5 himself & ! ,gov]nm5t8 ,if su* be ! actual cause ( ! delay1 7& we believe t1 9 "s degree1 x is s17 we _c refra9 f express+ our surprise t he does n adopt ! c&id & /raight=w>d c\rse ( decl>+ ! gr\nds ( 8 discont5t1 & expla9+ ! arrangem5t :i* he desires to be adopt$4 ,we d n hesitate to say1 t e reasonable accommodation & assi/ance ought to be af=d$ him4 ,b if he w p]t9aci\sly ab/a9 f ?is1 to our m9ds1 obvi\s & prop] c\rse1 !n x is surely ! duty ( ,gov]nm5t to appo9t prop] p]sons to enquire into & report on ! pres5t /ate ( ! ma*9]y2 to asc]ta9 ! causes ( its susp5sion2 & to recomm5d su* measures z may appe> to be mo/ effectual to ensure its spe$y completion4 ,if !y d n by su* means succe$ 9 putt+ ! project 9 a /ate ( advancem5t1 !y w at lea/ %ift f !mselves all responsibility = its susp5sion4 ,footnote #bc3 ,gov]nm5t has erect$ a fire-pro( build+1 9 :i* x is int5d$ t ! calculat+ ma*9]y %all be plac$ :5 complet$4 ,9 ?is build+ are n[ deposit$ ! l>ge collection ( draw+s1 conta9+ ! designs1 n only ( ! "p ( ! ma*9]y :i* has be5 already con/ruct$1 b :at is ( mu* great] importance1 ( ^? p>ts :i* h n yet be5 ev5 modell$4 ,x is gratify+ to "k t ,gov]nm5t has %[n a prop] solicitude = ! pres]vation ( ^? preci\s b p]i%able docum5ts1 ! loss or de/ruction ( :i* w\ld1 9 ! ev5t ( ! dea? ( ! inv5tor1 r5d] ! completion ( ! ma*9]y impracticable4