,,! ,,slide ,,rule3 ,a ,,practical ,,manual ,,by ,,*>les ,n4 ,,pickwor? ,,:itwor? ,,s*ol>2 ,,editor ,,( 8,,! ,,me*anical ,,_w02 ,,au?or ,,( 8,,log>i?ms ,,= ,,beg9n]s02 8,,! ,,indicator3 ,,its ,,con/ruction ,,& ,,application02 8,,! ,,indicator ,,diagram3 ,,its ,,analysis ,,& ,,calculation10 ,,etc4 ,,sev5te5? ,,edition ,,man*e/]3 ,,emmott ,,& ,,co41 ,,limit$1 #fe ,,k+ ,,/reet2 ,,new ,,york3 ,d4 ,,van ,,no/r& ,,co41 #h ,,w>r5 ,,/reet4 ,,london3 ,,emmott ,,& ,,co41 ,,limit$1 #bj ,,b$=d ,,/reet1 ,w4,c4 ,,& ,,pitman ,,& ,,sons1 ,,limit$1 ,,p>k] ,,/41 ,,k+sway1 ,w4,c4 #b4 ,?ree ,%ill+s & ,sixp5ce net ,all rights res]v$4 ,,preface ,,to ,,! ,,fifte5? ,,edition4 ,sev]al new slide rules = special calculations are describ$ 9 ?is edition1 & ! cont5ts fur!r ext5d$ to include a section deal+ ) screw-cutt+ ge> calculations by ! slide rule--an application ( ! in/rum5t to :i* att5tion has be5 giv5 rec5tly4 ,m5tion %\ld be made ( ! fact t "s ( ! special slide rules describ$ 9 previ\s editions are no long] obta9able4 ,z1 h[ev]1 ! descriptive notes may be ( s]vice to ^? possess+ ! in/rum5ts1 & are1 9 "s measure1 ( g5]al int]e/1 !y h be5 all[$ to rema9 9 ! pres5t issue4 ,! au?or t5d]s 8 ?anks to ! _m :o h ev9c$ _! appreciation ( 8 ef=ts to popul>ise ! subject2 also = ! _m k9d h9ts & sugge/ions :i* he has receiv$ f "t to "t1 & ) a cont9uance ( :i* he tru/s to be fav\r$ 9 ! future4 ,c4 ,n4 ,p4 ,,)+ton1 ,,man*e/]1 ,novemb] #aiag4 ,,preface ,,to ,,! ,,sev5te5? ,,edition4 ,! su/a9$ dem& = ?is v successful "w hav+ result$ 9 ! e>ly call = a new edition1 ! opportunity has be5 tak5 to introduce descriptions ( new slide rules & to effect "s slight revisions4 ,c4 ,n4 ,p4 ,,)+ton1 ,,man*e/]1 ,decemb] #aibj4 ,,cont5ts4 ,,page ,introductory #e ,! ,ma!matical ,pr9ciple ( ! ,slide ,rule #f ,notation by ,p[]s ( #aj #h ,! ,me*anical ,pr9ciple ( ! ,slide ,rule #i ,! ,primitive ,slide ,rule #aj ,! ,mod]n ,slide ,rule #ab ,! ,notation ( ! ,slide ,rule #ad ,! ,cursor or ,runn] #ag ,multiplication #ai ,division #bd ,! ,use ( ! ,upp] ,scales = ,multiplication & ,division #bf ,reciprocals #bg ,cont9u$ ,multiplication & ,division #bh ,multiplication & ,division ) ! ,slide ,inv]t$ #cj ,proportion #ca ,g5]al ,h9ts on ! ,elem5t>y ,uses ( ! ,slide ,rule #cf ,squ>es & ,squ>e ,roots #cg ,cubes & ,cube ,roots #dj ,miscellane\s ,p[]s & ,roots #de ,p[] & ,roots by ,log>i?ms #de ,o!r ,me?ods ( ,obta9+ ,p[]s & ,roots #dg ,comb9$ ,op]ations #di ,h9ts on ,evaluat+ ,expressions #eb ,gauge ,po9ts #ec ,examples 9 ,te*nical ,calculations #ef ,trigonometrical ,application #gd ,slide ,rules ) ,log-log ,scales #hd ,special ,types ( ,slide ,rules #ib ,long-,scale ,slide ,rules #if ,circul> ,calculators #aja ,slide ,rules = ,special ,calculations #aji ,con/ruction ,improvem5ts 9 ,slide ,rules #aaj ,! ,accuracy ( ,slide ,rule ,results #aaa ,app5dix3-- ,new ,slide ,rules #aac ,! ,solution ( ,algebraic ,equations #abb ,screw-,cutt+ ,ge> ,calculations #abd ,gauge ,po9ts & ,signs on ,slide ,rules #abf ,tables & ,data #abh ,slide ,rule ,data ,slips #acc ,,! ,,slide ,,rule4 ,,introductory4 ,! slide rule may be def9$ z an in/rum5t = me*anically effect+ calculations by log>i?ms4 ,^? famili> ) log>i?ms & _! use w recognise t ! slide rule provides :at is 9 effect a concisely arrang$ table ( log>i?ms1 toge!r ) a simple & conv5i5t means = add+ & subtract+ any select$ values4 ,^?1 h[ev]1 :o h no acqua9tance ) log>i?ms w f9d t only an elem5t>y k ( ! subject is necess>y to enable !m to make full use ( ! slide rule4 ,x is true t = simple slide-rule op]ations1 z multiplication & division1 a k ( log>i?ms is unnecess>y2 inde$1 _m :o h no consci\s und]/&+ ( log>i?ms make good use ( ! in/rum5t4 ,b ?is involves a bl9d reliance ^u rules )\t an appreciation ( _! orig9 or limitations1 & ?is1 9 turn1 eng5d]s a want ( confid5ce 9 ! results ( any b ! simple/ op]ations1 & prev5ts ! fulle/ use be+ made ( ! in/rum5t4 ,= ?is reason a brief1 b probably suffici5t rsum ( ! pr9ciples ( log>i?mic calculation w be giv54 ,^? desir+ a m detail$ explanation are ref]r$ to ! writ]'s 8,log>i?ms = ,beg9n]s40 ,! slide rule enables v>i\s ari?metical1 algebraical & trigonometrical processes to be p]=m$ ) ease & rapidity1 & ) suffici5t accuracy = mo/ practical purposes4 ,a grasp ( ! simple fundam5tal pr9ciples :i* und]lie its op]ation1 toge!r ) a little pati5t practice1 are all t are necess>y to acquire facility 9 us+ ! in/rum5t1 & few :o h become pr(ici5t 9 ?is sy/em ( calculat+ w\ld will+ly rev]t to ! labori\s ari?metical processes4 ,,! ,,ma!matical ,,pr9ciple ,,( ,,! ,,slide ,,rule4 ,log>i?ms may be def9$ z a s]ies ( numb]s 9 ari?metical progression1 z #j1 #a1 #b1 #c1 #d1 etc41 :i* be> a def9ite relation%ip to ano!r s]ies ( numb]s 9 geometrical progression1 z #a1 #b1 #d1 #h1 #af1 etc4 ,a m precise def9ition is3--,! log>i?m ( a numb] to any base1 is ! index ( ! p[] to :i* ! base mu/ be rais$ to equal ! giv5 numb]4 ,9 ! log>i?ms 9 g5]al use1 kn[n z common log>i?ms1 & ) :i* we are alone conc]n$1 #aj is ! base select$4 ,! g5]al def9ition may !re=e be /at$ 9 ! foll[+ modifi$ =m3--,! common log>i?m ( a numb] is ! index ( ! p[] to :i* #aj mu/ be rais$ to equal ! giv5 numb]4 ,apply+ ?is rule to a simple case1 z #ajj #ajb1 we see t ! base #aj mu/ be squ>$ 7i4e41 rais$ to ! #bnd p[]7 9 ord] to equal #ajj1 ! numb] select$4 ,!re=e1 z #b is ! index ( ! p[] to :i* #aj mu/ be rais$ to equal #ajj1 x foll[s f our def9ition t #b is ! common log>i?m ( #ajj4 ,simil>ly ! common log>i?m ( #ajjj w be #c1 :ile proce$+ 9 ! opposite direction ! common log4 ( #aj mu/ equal #a4 ,tabulat+ ^! results & ext5d+1 we h3-- ,numb]s #aj1jjj #ajjj #ajj #aj #a ,log>i?ms #d #c #b #a #j ,x w n[ be evid5t t = numb]s betwe5 #a & #aj ! logs4 w be betwe5 #j & #a #aj #ajj #a #b #ajj #ajjj #b #c #ajjj #aj1jjj #c #d ,9 o!r words1 ! log>i?ms ( numb]s betwe5 #a & #aj w be :olly fractional 7i4e41 decimal72 ! logs4 ( numb]s betwe5 #aj & #ajj w be #a foll[$ by a decimal quantity2 ! logs4 ( numb]s betwe5 #ajj & #ajjj w be #b foll[$ by a decimal quantity1 & s on4 ,^! decimal quantities = numb]s f #a to #aj 7:i* are ! log>i?ms ( ?is p>ticul> s]ies7 are z foll[s3-- ,numb]s #a #b #c #d #e #f #g #h #i #aj ,log>i?ms #j #jcja #jdgg #jfjb #jfii #jggh #jhde #jijc #jied #ajjj ,comb9+ ! two tables1 we c complete ! log>i?ms4 ,?us = #c multipli$ successively by #aj1 we h3-- ,numb]s #c #cj #cjj #cjjj #cj1jjj etc4 ,log>i?ms #jdgg #adgg #bdgg #cdgg #ddgg ,we see f ?is t = numb]s hav+ ! same significant figure 7or figures71 #c 9 ?is case1 ! decimal "p or mantissa ( ! log>i?m is ! same1 b t ! integral "p or *>act]i/ic is always "o less ?an ! numb] ( figures be=e ! decimal po9t4 ,= numb]s less ?an #a ! same plan is foll[$4 ,?us ext5d+ our fir/ table d[nw>ds1 we h3-- ,numb]s #a #ja #jja #jjja #jjjja etc4 ,log>i?ms #j #a #b #c #d s t = #c divid$ successively by #aj1 we h3-- ,numb]s #c #jc #jjc #jjjc #jjjjc etc4 ,log>i?ms #jdgg #adgg #bdgg #cdgg #ddgg ,"h aga9 we see t ) ! same significant figures 9 ! numb]s1 ! mantissa ( ! log>i?m has always ! same 7positive7 value1 b ! *>act]i/ic is "o m ?an ! numb] ( #j's imm$iately foll[+ ! decimal po9t1 & is negative1 z indicat$ by ! m9us sign writt5 ov] x4 ,only ! decimal p>ts ( ! log>i?ms ( numb]s betwe5 #a & #aj are giv5 9 ! usual tables1 =1 z %[n above1 ! log>i?ms ( all t5fold multiples or submultiples ( a numb] c be obta9$ at once by modify+ ! *>act]i/ic 9 accordance ) ! rules giv54 ,an exam9ation ( ! two r[s ( figures giv+ ! log>i?ms ( numb]s f #a to #aj w reveal "s /rik+ peculi>ities1 & at ! same "t s]ve to illu/rate ! pr9ciple ( log>i?mic calculation4 ,fir/1 x w be notic$ t ! addition ( any two ( ! log>i?ms gives ! log>i?m ( ! product ( ^! two numb]s4 ,?us1 ! addition ( log4 #b & log4 #d #jcja #jfjb #jijc1 & ?is is se5 to be ! log>i?m ( #h1 t is1 ( #b #d4 ,conv]sely1 ! diff]5ce ( ! log>i?ms ( two numb]s gives ! log>i?m ( ! quoti5t result+ f ! division ( ^! two numb]s4 ,?us1 log4 #h log4 #b #jijc #jcja #jfjb1 :i* is ! log4 ( #d1 or ( #h #b4 ,"o o!r important po9t is to be not$4 ,if ! log>i?m ( any numb] is multipli$ by #b1 #c1 or any o!r quantity1 :ole or fractional1 ! result is ! log>i?m ( ! orig9al numb]1 rais$ to ! #bnd1 #crd1 or o!r p[] respectively4 ,?us1 multiply+ ! log4 ( #c by #b1 we obta9 #jdgg #b #jied1 & ?is is se5 to be ! log4 ( #i1 t is1 ( #c rais$ to ! #bnd p[]1 or #c squ>$4 ,aga91 log4 #b multipli$ by #c #jijc--t is1 ! log4 ( #h1 or ( #b rais$ to ! #crd p[]1 or #b cub$4 ,conv]sely1 divid+ ! log>i?m ( any orig9al numb] by any numb] n1 we obta9 ! log>i?m ( ! n? root ( ! orig9al numb]4 ,?us1 log4 #h #c #jijc #c #jcja1 & is !re=e equal to log4 #b or to ! log4 ( ! cube root ( #h4 ,only simple logs4 h be5 tak5 9 ^! examples1 b ! /ud5t w und]/& t ! same reason+ applies1 :atev] ! numb]4 ,?us = #bjc we prefix ! *>act]i/ic 7#a 9 ?is case7 to log4 #b1 giv+ #acja4 ,multiply+ by #c1 we h #cijc z ! result+ log>i?m1 & z its *>act]i/ic is #c1 we "k t x corresponds to ! numb] #hjjj4 ,h5ce #bjc #hjjj4 ,9 ?is brief explanation is includ$ all t ne$ n[ be said ) reg>d to ! prop]ties ( log>i?ms4 ,! ma9 facts to be borne cle>ly 9 m9d are3--7#a47 ,t to f9d ! product ( two numb]s1 ! log>i?ms ( ! numb]s are to be add$ toge!r1 ! result be+ ! log>i?m ( ! product requir$1 ! value ( :i* c !n be det]m9$4 7#b47 ,t 9 f9d+ ! quoti5t result+ f ! division ( "o numb] by ano!r1 ! diff]5ce ( ! log>i?ms ( ! numb]s gives ! log>i?m ( ! quoti5t1 f :i* ! value ( ! latt] c be asc]ta9$4 7#c47 ,t to f9d ! result ( rais+ a numb] to ! n? p[]1 we multiply ! log>i?m ( ! numb] by n1 ?us obta9+ ! log>i?m1 & h5ce ! value1 ( ! desir$ result4 ,& 7#d47 ,t to f9d ! n? root ( a numb]1 we divide ! log>i?m ( ! numb] by n1 ?is giv+ ! log>i?m ( ! result1 f :i* its value may be det]m9$4 ,,notation ,,by ,,p[]s ,,( #aj4 ,a conv5i5t me?od ( repres5t+ an ari?metical quantity is to split x up into two factors1 ( :i* ! fir/ is ! orig9al numb]1 ) ! decimal po9t mov$ s z to imm$iately foll[ ! fir/ significant figure1 & ! second1 #ajn :]e n is ! numb] ( places ! decimal po9t has be5 mov$1 ?is index be+ positive = numb]s great] ?an #a1 & negative = numb]s less ?an #a4a ,9 ?is sy/em1 !re=e1 we reg>d #c1faj1jjj z #cfa #a1jjj1jjj1 & write x z #cfa #ajf4 ,simil>ly #cfa #cfa x #ajb2 #jjcfa 7 7#cfa7_/7ajj77 #cfa #ajb2 #jjjjjcfa #cfa #aje1 etc4 ,to re/ore a numb] to its orig9al =m1 we h only to move ! decimal po9t ?r\gh ! numb] ( places indicat$ by ! index1 mov+ to ! "r if ! index is positive & to ! left 7prefix+ #j's7 if negative4 ,?is me?od1 :i* %\ld be cultivat$ = ord9>y ari?metical "w1 is sub/antially t foll[$ 9 calculat+ by ! slide rule4 ,?us ) ! slide rule ! multiplication ( #fc1bjj by #jjjce virtually resolves itself into #fcb #ajd #ce #ajc or #fcb #ce #ajd-c #bbab x #aja #bbab4 ,x w be se5 lat]1 h[ev]1 t ! result c be arriv$ at by a m direct1 if less sy/ematic1 me?od ( work+4 ,,! ,,me*anical ,,pr9ciple ,,( ,,! ,,slide ,,rule4 ,illu/ration3 ,,fig4 #a4 ,! me*anical pr9ciple involv$ 9 ! slide rule is ( a v simple "*4 ,9 ,fig4 #a1 ,a & ,b repres5t two rules divid$ into #aj equal p>ts1 ! division l9es be+ numb]$ consecutively z %[n4 ,if ! rule ,b is mov$ to ! "r until #j on ,b is opposite #c on ,a1 x is se5 t any numb] on ,a is equal to ! co9cid+ numb] on ,b1 plus #c4 ,?us opposite #d on ,b is #g on ,a4 ,! reason is obvi\s4 ,by mov+ ,b to ! "r1 we add to a l5g? #jc1 ano!r l5g? #jd1 ! result read (f on ,a be+ #g4 ,evid5tly1 ! same result w\ld h be5 obta9$ if a l5g? #jd _h be5 add$1 by means ( a pair ( divid]s1 to ! l5g? #jc on ! scale ,a4 ,by means ( ! slide ,b1 h[ev]1 ! addition is m readily effect$1 &1 :at is ( mu* great] importance1 ! result ( add+ #c to any "o ( ! numb]s )9 range1 on ! l[] scale1 is imm$iately se5 by read+ ! adjac5t numb] on ,a4 ,( c\rse1 subtraction c be q z readily p]=m$4 ,?us1 to subtract #d f #g1 we require to d$uct f #jg on ! ,a scale1 a l5g? #jd on ,b4 ,we d ?is by plac+ #d on ,b "u #g on ,a1 :5 ov] #j on ,b we f9d #c1 on ,a4 ,x is "h evid5t t ! diff]5ce ( any pair ( co9cid+ numb]s on ! scales is con/antly equal to #c4 ,illu/ration3 ,,fig4 #b4 ,an important modification results if ! slide-scale ,b is inv]t$ z 9 ,fig4 #b4 ,9 ?is case1 to f9d ! sum ( #d & #c we require to place ! #d ( ! ,a scale to #c on ! ,b scale1 & ! result is read on ,a ov] #j on ,b4 ,"h x w be not$1 ! sum ( any pair ( co9cid+ numb]s on ! scales is con/ant & equal to #g4 ,?is case1 !re=e1 resembles t ( ! imm$iately prec$+ "o1 except t ! sum1 in/ead ( ! diff]5ce1 ( any pair ( co9cid+ numb]s is con/ant4 ,to f9d ! diff]5ce ( two factors1 ! conv]se op]ation is necess>y4 ,?us1 to subtract #d f #g1 #j on ,b is plac$ opposite #g on ,a1 & ov] #d on ,b is f\nd #c on ,a4 ,f ^! examples x w be se5 t ) ! slide inv]t$ ! me?ods ( op]ation are ! rev]se ( ^? us$ :5 ! slide is 9 its normal position4 ,x w be und]/ood t al?\gh we h only consid]$ ! prim>y divisions ( ! scales1 ! rem>ks apply equally to any subdivisions into :i* ! prim>y spaces ( ! scales might be divid$4 ,fur!r1 we note t ! l5g? ( scale tak5 to repres5t a unit is q arbitr>y4 ,,! ,,primitive ,,slide ,,rule4 ,! application ( ! =ego+ pr9ciples to ! slide rule c be %[n mo/ conv5i5tly by describ+ ! con/ruction ( a simple =m ( slide rule3--,take a /rip ( c>d ab\t #aa 94 long & #b 94 wide2 draw a l9e d[n ! c5tre ( its wid?1 & m>k (f two po9ts1 #aj 94 ap>t4 ,draw cross l9es at ^! po9ts & figure !m #a & #aj on ea* side1 z 9 ,fig4 #c4 ,next m>k (f l5g?s ( #cja1 #dgg1 #fjb1 #fii1 #ggh1 #hde1 #ijc & #ied in*es1 f ! l9e m>k$ #a4 ,draw cross l9es z be=e1 & figure ^! l9es1 #b1 #c1 #d1 #e1 #f1 #g1 #h & #i4 ,to fill 9 ! int]m$iate divisions ( ! scale1 take ! logs1 ( #aa1 #ab1 #ac1 etc4 7f a table71 multiply ea* by #aj1 & ?us obta9 ! di/ances f #a1 at :i* ! sev]al subdivisions are to be plac$4 ,m>k ^! #ab1 #ac1 #ad1 etc41 & complete ! scale1 mak+ ! int]polat$ division m>ks %ort] to facilitate read+1 z ) an ord9>y measur+ rule4 ,cutt+ ! c>d cleanly d[n ! c5tre l9e1 we h ! ess5tials ( ! slide rule4 ,illu/ration3 ,,fig4 #c4 ,! fundam5tal pr9ciple ( ! slide rule is n[ evid5t3--,ea* scale is graduat$ 9 su* a mann] t ! di/ance ( any numb] f #a is proportional to ! log>i?m ( t numb]4 ,illu/ration3 ,,fig4 #d4 8,we "k t to f9d ! product ( #b #c by log>i?ms1 we add #jcja1 or log4 #b1 to #jdgg1 ! log4 ( #c1 obta9+ #jggh1 or log4 #f4 ,) our primitive slide rule we place #a on ! l[] scale to #cja 94 7:i* we h m>k$ #b7 on ! upp] scale 7,fig4 #d74 ,!n ov] #dgg 94 on ! l[] scale 7:i* we m>k$ #c71 we h #ggh 94 7:i* we m>k$ #f7 on ! upp] scale4 ,conv]sely1 to divide #f by #c1 we place #c on ! l[] scale 9 agreem5t ) #f on ! upp]1 & ov] #a on ! l[] scale read #b on ! upp] scale4 ,?is me?od ( add+ & subtract+ scale l5g?s w be se5 to be id5tical ) t us$ 9 ! simple case %[n 9 ,fig4 #a4 ,,! ,,mod]n ,,slide ,,rule4 ,! mod]n =m ( slide rule1 v>i\sly /yl$ ! ,gravt1 ! ,tav]ni]-,gravt1 & ! ,mannheim rule1 is frequ5tly made ( boxwood1 b all ! lead+ in/rum5t mak]s n[ supply rules made ( boxwood or mahogany1 & fac$ ) celluloid1 ! :ite surface ( :i* br+s out ! graduations mu* m di/9ctly ?an l9es engrav$ on a boxwood surface4 ,! celluloid fac+s %\ld n be poli%$1 z a dull surface is mu* less fatigu+ to ! eyes4 ,! mo/ g5]ally us$1 & on ! :ole ! mo/ conv5i5t size ( rule1 is ab\t #aj;94 long1 #a;94 wide1 & ab\t 94 ?ick2 b #e 941 #h 941 #ae 941 #bj 941 #bd 94 & #dj 94 rules are also made4 ,9 ! c5tre ( ! /ock ( ! rule a movable slip is fitt$1 :i* con/itutes ! slide1 & corresponds to ! l[] ( ! two rules ( our rudim5t>y examples4 ,illu/ration3 ,,fig4 #e4 ,f ,fig4 #e1 :i* is a repres5tation ( ! face ( a ,gravt or ,mannheim slide rule1 x w be se5 t f\r s]ies ( log>i?mic graduations or scale-l9es are employ$1 ! upp] & l[] be+ engrav$ on ! /ock or body ( ! rule1 :ile ! o!r two are engrav$ ^u ! slide4 ,! two upp] sets ( graduations are exactly alike 9 e p>ticul>1 & ! l[] sets are also simil>4 ,x is usual to id5tify ! two upp] scale-l9es by ! lett]s ,a & ,b1 & ! two l[] by ! lett]s ,c & ,d1 z indicat$ 9 ! figure at ! left-h& extremities ( ! scales4 ,ref]r+ to ! scales ,c & ,d1 ^! w ea* be se5 to be a developm5t ( ! elem5t>y scales ( ,fig4 #c1 b 9 ?is case ea* pr9cipal space is subdivid$1 m or less m9utely4 ,! pr9ciple1 h[ev]1 is exactly ! same1 s t by mov+ ! slide 7c>ry+ scale ,c71 multiplication & division c be me*anically p]=m$ 9 ! mann] describ$4 ,! upp] scale-l9e ,a consi/s ( two exactly simil> scales1 plac$ end to end1 ! fir/ ly+ betwe5 ,,il & ,,ic1 & ! second betwe5 ,,ic & ,,ir4 ,! fir/ ( ^! scales w be designat$ ! left-h& ,a scale1 & ! second ! "r-h& ,a scale4 ,simil>ly ! co9cid+ scales on ! slide are ! left-h& ,b scale & ! "r-h& ,b scale4 ,ea* ( ^! f\r scales is divid$ 7z f9ely z conv5i5t7 z 9 ! case ( ! ,c & ,d scales1 b1 ( c\rse1 !y are exactly "o half ! l5g? ( ! latt]4 ,! two end graduations ( bo? ! ,c & ,d scales are kn[n z ! left- & "r-h& indices ( ^! scales4 ,sometimes !y are figur$ #a & #aj respectively2 sometimes bo? are m>k$ #a4 ,simil>ly ,,il & ,,ir are ! left- & "r-h& indices ( ! ,a & ,b l9es1 :ile ,,ic is ! c5tre index ( ^! scales4 ,o!r division l9es usually f\nd on ! face ( ! rule are "o on ! left-h& ,a & ,b scales1 indicat+ ! ratio ( ! circumf]5ce ( a circle to its diamet]1 #cadaf2 & a l9e on ! "r-h& ,b scale m>k+ ! position ( 77_/7#d7 #jghed1 us$ 9 calculat+ ! areas ( circles4 ,ref]5ce w be made h]eaft] to ! scales on ! "u-side ( ! slide1 & we ne$ n[ only add t "o ( ! edges ( ! rule1 usually bevell$1 is g5]ally graduat$ 9 millimetres1 :ile ! o!r edge has engrav$ on x a scale ( in*es divid$ into eigh?s or t5?s4 ,on ! bottom face inside ! groove ( ! rule ei!r "o or ! o!r ( ^! scales is cont9u$ 9 su* a mann] t by draw+ ! slide out to ! "r & us+ ! scale inside ! rule1 9 conjunction ) ! correspond+ scale on ! edge1 x is possible to measure #bj in*es 9 ! "o case1 or ne>ly #ejj millimetres 9 ! o!r4 ,on ! back ( ! rule !re is usually a collection ( data1 = :i* ! slips giv5 at ! end ( ?is "w may (t5 be sub/itut$ ) advantage4 ,,! ,,notation ,,( ,,! ,,slide ,,rule4 ,hi!rto our att5tion has be5 conf9$ to a consid]ation ( ! prim>y divisions ( ! scales4 ,! same pr9ciple ( graduation is1 h[ev]1 us$ ?r\gh\t2 & aft] :at has be5 said1 ?is "p ( ! subject ne$ n be fur!r enl>g$ ^u4 ,"s explanation ( ! me?od ( read+ ! scales is necess>y1 z facility 9 us+ ! in/rum5t dep5ds 9 a v great measure ^u ! dext]ity ( ! op]ator 9 assign+ ! correct value to ea* division on ! rule4 ,by ref]5ce to ,fig4 #e1 x w be se5 t ea* ( ! prim>y spac+s 9 ! sev]al scales is inv>iably subdivid$ into t52 b s9ce ! l5g?s ( ! successive prim>y divisions rapidly dim9i%1 x is impossible to subdivide ea* ma9 space into ! same numb] ( p>ts t ! space #a-b c be subdivid$4 ,?is v>iable spac+ ( ! scales is at fir/ confus+ to ! /ud5t1 b ) a little practice ! difficulty is soon ov]come4 ,) ! ,c or ,d scale1 x w be notic$ t ! l5g? ( ! int]val #a-b is suffici5t to all[ ea* ( ! #aj subdivisions to be aga9 divid$ into #aj p>ts1 s t ! :ole int]val #a-b is divid$ into #ajj4 ,! %ort] ma9 space #b-c1 & ! /ill %ort] "o #c-d1 only all[ ( ! #aj subdivisions ( ea* be+ divid$ into five p>ts4 ,ea* ( ^! ma9 spaces is !re=e divid$ into #ej p>ts4 ,= ! rema9d] ( ! scale ea* ( ! #aj subdivisions ( ea* ma9 space is divid$ into two p>ts only2 s t f ! ma9 division #d to ! end ( ! scale ! prim>y spaces are divid$ into #bj p>ts only4 ,9 ! upp] scales ,a or ,b1 x w be f\nd t--z ! space #a-b is ( only half ! l5g? ( ! correspond+ space on ,c or ,d--! #aj subdivisions ( ?is int]val are divid$ into five p>ts only4 ,simil>ly ea* ( ! #aj subdivisions ( ! int]vals #b-c1 #c-d1 & #d-e are fur!r divid$ into two p>ts only1 :ile = ! rema9d] ( ! scale only ! #aj subdivisions are possible1 ow+ to ! rapidly dim9i%+ l5g?s ( ! prim>y spac+s4 ,! values actually giv5 on ! rule run f #a to #aj on ! l[] scales & f #a to #ajj on ! upp] scales1 &1 z expla9$ on page #i1 all factors are br\ght )9 ^! ranges ( values by multiply+ or divid+ !m by p[]s ( #aj4 ,by foll[+ ?is plan1 we virtually reg>d ea* factor z m]ely a s]ies ( significant figures1 & make ! necess>y modification due to ! 8p[]s ( #aj0 :5 fix+ ! position ( ! decimal po9t 9 ! answ]4 ,_m1 h[ev]1 f9d x conv5i5t 9 practice to reg>d ! values on ! rule z multipli$ or divid$ by su* p[]s ( #aj z may be necess>y to suit ! factors ent]+ into ! calculation4 ,if ?is plan is adopt$1 ! values giv5 to ea* graduation ( ! scales w dep5d on t giv5 to ! left index figure 7#a7 ( ! l[] scales1 ?is be+ any multiple or submultiple ( #aj4 ,?us ,,il on ! ,d scale may be reg>d$ z #a1 #aj1 #ajj1 #ajjj1 etc41 or z #ja1 #jja1 #jjja1 #jjjja1 etc42 b once ! initial value is assign$ to ! index1 ! ratio ( value mu/ be ma9ta9$ ?r\gh\t ! :ole scale4 ,= example1 if #a on ,c is tak5 to repres5t #aj1 ! ma9 divisions #b1 #c1 #d1 etc41 w be read z #bj1 #cj1 #dj1 etc4 ,on ! o!r h&1 if ! f\r? ma9 division is read z #jjjd1 !n ! left index figure ( ! scale w be read z #jjja4 ,! figur$ subdivisions ( ! ma9 space #a-b are to be read z #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ah & #ai--;if ! index repres5ts #aj1--;& z correspond+ multiples = any o!r value ( ! index4 ,indep5d5tly consid]$1 ^! rem>ks apply equally to ! ,a or ,b scale1 b 9 ?is case ! notation is cont9u$ ?r\gh ! second half ( ! scale1 ! figures ( :i* are to be read z t5fold values ( ! correspond+ figures 9 ! fir/ half ( ! scale4 ,! read+ ( ! int]m$iate divisions w1 ( c\rse1 be det]m9$ by ! values assign$ to ! ma9 divisions4 ,?us1 if ,,il on ,d is read z #a1 !n ea* ( ! smalle/ subdivisions ( ! space #a-b w be read z #jja1 & ea* ( ! smalle/ subdivisions ( ! spaces #b-c or #c-d z #jjb1 :ile = ! rema9d] ( ! scale ! smalle/ subdivisions are read z #jje4 ,9 ! ,a or ,b scale ! subdivisions ( ! space #a-b ( ! fir/ half ( ! scale are 7if ,,il #a7 read z #jjb1 #jjd1 etc42 = ! divisions #b-c1 #c-d1 & #d-e1 ! smalle/ int]vals are read z #jje ( ! prim>y spaces1 & f #e to ! c5tre index ( ! scale ! divisions repres5t #ja ( ea* ma9 int]val4 ,pass+ ! c5tre index1 :i* is1 n[ read z #aj1 ! smalle/ subdivisions imm$iately foll[+ are read #ajb1 #ajd1 etc41 until #bjj is rea*$2 !n we read #bje1 #baj1 #bae #bbj1 etc41 until ! figur$ ma9 division #e is rea*$4 ,! rema9d] ( ! scale is read #ea1 #eb1 #ec1 etc41 up to #ajj1 ! "r-h& index4 ,fur!r subdivision ( any ( ! spaces ( ! rule c be effect$ by ! eye1 & aft] a little practice ! op]ator w become q exp]t 9 e/imat+ any int]m$iate value4 ,x af=ds good practice to set #a on ,c to #ajd1 #aji1 etc4 on ,d1 & to read ! values on ,d1 "u #d1 #f1 #h1 etc4 on ,c4 ,z ! exact results are easily calculat$ m5tally1 ! /ud5t1 by ?is means1 w receive bett] in/ruction 9 e/imat+ int]m$iate results ?an c be giv5 by any diagram4 ,"s rules w be f\nd figur$ z %[n 9 ,fig4 #e2 9 o!rs1 ! "r-h& upp] scales are m>k$ #aj1 #bj1 #cj1 etc4 ,aga91 o!rs are m>k$ decimally1 ! l[] scales & ! left-h& upp] scales be+ figur$ #a1 #aa1 #ab1 #ac 444 #be1 etc4 ,! latt] =m has advantages f ! po9t ( view ( ! beg9n]4 ,! me?od ( read+ ! ,a & ,b scales1 j giv51 applies only :5 ^! scales are reg>d$ z altoge!r indep5d5t ( ! l[] pair ( scales ,c & ,d4 ,"s op]ators pref] to use ! ,a & ,b scales1 & "s ! ,c & ,d scales1 = ! ord9>y op]ations ( proportion1 multiplication1 & division4 ,ea* me?od has its advantages1 z w be %[n1 b 9 ! m complex calculations1 z involution & evolution1 etc41 ! relation ( ! upp] scales to ! l[] scales becomes a v important factor4 ,! di/ance #a-aj on ! upp] scales is "o-half ( ! di/ance #a-aj on ! l[] scales4 ,h5ce any di/ance f #a1 tak5 on ! upp] scales1 repres5ts twice ! log>i?m :i* ! same di/ance repres5ts on ! l[] scales4 ,9 o!r words1 ! l5g? :i* repres5ts log4 ,n on ,d1 w\ld repres5t #b log4 ,n on ,a2 &1 conv]sely1 ! l5g? :i* repres5ts log4 ,n on ,a1 w\ld repres5t 7log4 ,n7_/7#b7 on ,d4 ,n[ we h se5 7page #h7 t multiply+ ! log4 ( a numb] by #b gives ! log4 ( ! squ>e ( ! numb]4 ,h5ce1 above any numb] on ,d we f9d its squ>e on ,a1 or1 conv]sely1 bel[ any numb] on ,a1 we f9d its squ>e root on ,d4 ,?us1 above #b we f9d #d2 "u #di1 we f9d #g & s on4 ,obvi\sly ! same relation exi/s betwe5 ! ,b & ,c scales4 ,,! ,,cursor ,,or ,,runn]4 ,all mod]n slide rules are n[ fitt$ ) a cursor or runn]1 :i* usually consi/s ( a light metal frame mov+ "u spr+ control 9 grooves 9 ! edges ( ! /ock ( ! rule4 ,?is frame c>ries a piece ( glass1 mica or transp>5t celluloid1 ab\t #a 94 squ>e1 across ! c5tre ( :i* a f9e ref]5ce l9e is drawn exactly at "r angles to ! l9e ( scales4 ,to 8set ! cursor0 to any value on ! scales ( ! rule1 ! frame is tak5 betwe5 ! ?umb & =ef+] & adju/$ 9 position until ! l9e falls exactly ^u ! graduation1 or ^u an e/imat$ value1 betwe5 a pair ( graduations1 z ! case may be4 ,hav+ fix$ "o numb] 9 ?is way1 ano!r value on ei!r ( ! scales on ! slide may be simil>ly adju/$ 9 ref]5ce to ! cursor l9e4 ,! cursor w be f\nd v conv5i5t 9 mak+ su* sett+s1 especially :5 ei!r or bo? ( ! numb]s are locat$ by eye e/imation4 ,x also f9ds a v important use 9 ref]r+ ! read+s ( ! upp] scale to ^? ( ! l[]1 or vice v]sa1 :ile z an aid 9 cont9u$ multiplication & division & complex calculations g5]ally1 its value is ine/imable4 ,multiple ,l9e ,cursors4--,cursors c be obta9$ ) two l9es1 ! di/ance betwe5 !m be+ t betwe5 #ghed & #aj on ! ,a scale4 ,! use ( ?is cursor is expla9$ on page #eg4 ,ano!r multiple l9e cursor has %ort l9es engrav$ on x1 correspond+ to ! ma9 graduations f #ie to #aje on ! respective scales4 ,?is is useful = add+ or d$uct+ small p]c5tages4 ,! ,brok5 ,l9e ,cursor4--,to facilitate sett+1 brok5 l9e cursors are made1 9 :i* ! hair-l9e is n cont9u$ across ! scales1 b has two gaps1 z %[n 9 ,fig4 #f4 ,! ,po9t$ ,cursor has an index or po9t]1 ext5d+ ov] ! bevell$ edge ( ! rule1 on :i* is a scale ( in*es4 ,x is useful = summ+ ! l5g?s ( ! ord9ates ( indicator diagrams1 & also = plott+ l5g?s repres5t+ ! log>i?ms ( numb]s1 sometimes requir$ 9 graphic calculations4 ,! ,g\ld+ ,cursor4--,x has be5 po9t$ out t 9 ord] to obta9 ! ?ird or f\r? figure ( a read+ on ! #aj 94 slide rule1 x is frequ5tly necess>y to dep5d ^u ! op]ator's ability to m5tally subdivide ! space )9 :i* ! read+ falls4 ,?is subdivision c be me*anically effect$ by ! aid ( ! ,g\ld+ ,cursor 7,fig4 #g71 :i* consi/s ( a frame fitt+ into ! usual grooves 9 ! rule1 & c>ry+ a metal plate fac$ ) celluloid1 ^u :i* is engrav$ a triangul> scale ,a ,b ,c4 ,! portion c>ry+ ! *isel edges ,e is n fix$ to ! cursor prop]1 b slides on ! latt]1 s t ! index m>ks on ! project+ prongs c be mov$ slightly along ! scales ( ! rule1 ?is movem5t be+ effect$ by ! %ort end ( ! b5t lev] ,f work+ 9 ! slot z %[n4 ,d is a po9t] :i* c be mov$ along ,f "u spr+ control4 ,z illu/rat+ ! me?od ( use1 we w assume t #a on ,c is plac$ to #aee on ,d1 & t we require to read ! value on ,d "u #bg on ,c4 ,?is is se5 to lie betwe5 #daej & #dbjj1 s sett+ ! po9t] ,d to ! l9e ,b ,c--always ! fir/ op]ation--we move ! :ole along ! rule until ! index l9e on ! l[] prong agrees ) #dbjj4 ,we !n move ,f across ! scale until ! index l9e agrees ) #dajj1 set ! po9t] ,d to ! l9e ,a ,c1 & move ! lev] back until ! index l9e agrees ) #bg on ! slide4 ,x w !n be f\nd t ! po9t] ,d gives #he on ,a ,b z ! value ( ! supplem5t>y figures1 & h5ce ! complete read+ is #dahe4 ,illu/ration3 ,,fig4 #f4 ,illu/ration3 ,,fig4 #h4 ,illu/ration3 ,,fig4 #g4 ,illu/ration3 ,,fig4 #i4 ,magnify+ ,cursors are ( assi/ance 9 read+ ! scales1 & 9 a good & direct light are v helpful4 ,9 "o =m an ord9>y l5s is c>ri$ by two light arms h+$ to ! upp] & l[] edges ( ! cursor1 s t x c be fold$ d[n to ! face ( ! rule :5 n 9 use4 ,a m compact =m1 %[n 9 ,fig4 #h1 consi/s ( a /rip ( plano-convex glass1 on ! "u-side ( :i* is ! hair-l9e4 ,9 a cursor made by ,ne/l] ( ,lahr1 ! plano-convex /rip is fix$ on ! ord9>y cursor4 ,! magnify+ p[] is ab\t #b1 s t a #e 94 rule1 hav+ ! same numb] ( graduations z a #aj 94 rule1 c be read ) equal facility1 by ! aid ( ?is cursor4 ,! ,digit-regi/]+ ,cursor1 suppli$ by ,mr4 ,a4 ,w4 ,fab]1 ,london1 & %[n 9 ,fig4 #i1 has a semicircul> scale runn+ f #j at ! c5tre upw>d to #f & d[nw>d to #f4 ,a small f+] enables ! op]ator to regi/] ! numb] ( digits to be add$ or subtract$ at ! end ( a l5g?y op]ation1 z expla9$ at page #bh4 ,,multiplication4 ,9 ! prelim9>y notes x 0 %[n t by me*anically add+ two l5g?s repres5t+ ! log>i?ms ( two numb]s1 we c obta9 ! product ( ^! numb]s2 :ile by subtract+ "o log4 l5g? f ano!r1 ! numb] repres5t$ by ! latt] is divid$ by ! numb] repres5t$ by ! =m]4 ,h5ce1 us+ ! ,c & ,d scales1 we h ! ,,rule ,,= ,,multiplication4--,set ! index ( ! ,c scale to "o ( ! factors on ,d1 & "u ! o!r factor on ,c1 f9d ! product on ,d4 ,illu/ration3 ,,fig4 #aj4 ,?us1 to f9d ! product ( #b #d1 ! slide is mov$ to ! "r until ! left index 7#a7 ( ,c is br\ght ov] #b on ,d1 :5 "u ! o!r factor 7#d7 on ,c1 is f\nd ! requir$ product 7#h7 on ,d4 ,foll[+ along ! slide1 to ! "r1 we f9d t beyond #e on ,c 7giv+ #aj on ,d71 we h no scale bel[ ! project+ slide 7,fig4 #aj74 ,if we imag9e ! ,d scale prolong$ to ! "r1 we %\ld h a repetition ( ! e>li] portion1 b1 z ) ! two p>ts ( ! ,a scales1 ! repeat$ portion w\ld be ( t5fold value1 & #aj on ,c w\ld agree ) #bj on ! prolong$ ,d scale4 ,we turn ?is fact to acc\nt by mov+ ! slide to ! left until #aj on ,c agrees ) #b on ,d1 & we c !n read (f su* results z #b #f #ab2 #b #h #af1 etc41 rememb]+ t z ! scale is n[ ( t5fold value1 !re w be two figures 9 ! result4 ,h5ce1 = ^? :o pref] rules1 we h ! ,,rule ,,= ,,! ,,numb] ,,( ,,digits ,,9 ,a ,,product4--,if ! product is read ) ! slide project+ to ! ,,left1 ,,add ,,! ,,numb] ,,( ,,! ,,digits ,,9 ,,! ,,two ,,factors2 if read ) ! slide to ! ,,"r1 d$uct #a f ?is sum4 ,,ex4--#be #gj #agej4 ,! product is f\nd ) ! slide project+ to ! left1 s ! numb] ( digits 9 ! product #b #b #d4 ,,ex4--#cf #be #ij4 ,! slide projects to ! "r1 & ! numb] ( digits 9 ! product is !re=e #a #b #a #b4 ,,ex4--#jjbe #jg #jjage4 ,! product is obta9$ ) ! slide project+ to ! left1 & ! numb] ( digits is !re=e #a #j #a4 ,,ex4--#jjjjahd #jjje #jjjjjjjib4 ,! sum ( ! numb] ( digits 9 ! two factors #c 7#b7 #e1 b z ! slide projects to ! "r1 ! numb] ( digits w be #e #a #f4 ,f ! la/ two examples x w be se5 t :5 ! fir/ significant figure ( a decimal factor does n imm$iately foll[ ! decimal po9t1 ! m9us sign is to be prefix$ to ! numb] ( digits1 c\nt+ z _m digits m9us z !re are #j's foll[+ ! decimal po9t4 ,?us1 #jjc has #a digit1 #jjjce has #b digits1 & s on4 ,"s little c>e is necess>y to ensure ^! m9us values be+ correctly tak5 into acc\nt 9 det]m9+ ! numb] ( digits 9 ! answ]4 ,= ?is reason _m pref] to treat decimal factors z :ole numb]s1 & to locate ! decimal po9t accord+ to ! usual rules = ! multiplication ( decimals4 ,?us1 9 ! la/ example we take #ahd #e #ibj1 b z by ! usual rule ! product mu/ conta9 #f #c #i decimal places1 we prefix six cyph]s1 obta9+ #jjjjjjjib4 ,:5 bo? factors consi/ ( integ]s z well z decimals1 ! numb] ( digits 9 ! product1 & !re=e ! position ( ! decimal po9t1 w be det]m9$ by ! usual rule = :ole numb]s4 ,ano!r me?od ( det]m9+ ! numb] ( digits 9 a product des]ves m5tion1 :i*1 n be+ dep5d5t ^u ! position ( ! slide1 is applicable to all calculat+ in/rum5ts4 ,,g5]al ,,rule ,,= ,,numb] ,,( ,,digits ,,9 ,a ,,product4--,:5 ! fir/ significant figure 9 ! product is small] ?an 9 ,,ei!r ( ! factors1 ! numb] ( digits 9 ! product is equal to ! ,,sum ( ! digits 9 ! two factors4 ,:5 ! contr>y is ! case1 ! numb] ( digits is #a ,,less ?an ! sum ( ! digits 9 ! two factors4 ,:5 ! fir/ figures are ! same1 ^? foll[+ mu/ be comp>$4 ,e/imation ( ! ,figures 9 a ,product4--,we h giv5 rules = ^? :o pref] to decide ! numb] ( figures by ?is means1 b exp]i5ce w %[ t to make ! be/ use ( ! in/rum5t1 ! result1 z read on ! rule1 %\ld be reg>d$ m]ely z ! significant figures ( ! answ]1 ! position ( ! decimal po9t1 if n obvi\s1 be+ decid$ by a v r\gh m5tal calculation4 ,9 v _m in/ances1 ! magnitude ( ! result w be evid5t f ! conditions ( ! problem--e4g41 :e!r ! answ] %\ld be #jc 941 #c 941 or #cj 942 or #aj tons1 #ja ton1 #ajj tons1 etc4 ,9 ^? cases :]e ! magnitude ( ! answ] _c be e/imat$1 & ! factors conta9 _m figures1 or h a numb] ( #j's foll[+ ! decimal po9t1 ! use ( notation by p[]s ( #aj 7page #h7 is ( consid]able assi/ance2 b m usually x w be f\nd1 t a v r\gh calculation w settle ! po9t ) comp>atively little tr\ble4 ,consid]able practice is ne$$ to "w rapidly & ) c]ta9ty1 :5 us+ rules4 ,moreov]1 ! exp]i5ce ?us acquir$ is conf9$ to slide-rule "w4 ,! same "t sp5t 9 practis+ ! 8r\gh approximation0 me?od w enable reliable results to be obta9$ rapidly1 ) ! advantage t ! me?od is applicable to calculations g5]ally4 ,h[ev]1 ! *oice ( me?ods is a matt] ( p]sonal pref]5ce4 ,bo? me?ods w be giv51 b :i*ev] plan is foll[$1 ! /ud5t is /rongly advis$ to cultivate ! habit ( =m+ an idea ( ! magnitude ( ! result4 ,,ex4--#ccf #bcf #gicj4 ,sett+ #a on ,c to #ccf on ,d1 we read "u #bcf on ,d & f9d #gic on ,d1 z ! significant figures ( ! answ]4 ,a r\gh calculation1 z #cj #bjj #fjjj1 indicates t ! result w consi/ ( #d figures1 & is !re=e to be read z #gicj4 ,,ex4--#ag1cjj #cghj #fe1djj1jjj4 ,by factoris+ ) p[]s ( #aj #agc #ajd #cgh #ajc #agc #cgh #ajg4 ,sett+ #a on ,c to #agc on ,d1 we read1 "u #cgh on ,c1 ! result ( ! simple multiplication1 z #fed4 ,multiply+ by #ajg moves ! decimal po9t #g places to ! "r1 & ! answ] is #fe1djj1jjj4 ,if x is requir$ to f9d a s]ies ( products ( :i* "o ( ! factors is con/ant1 set #a on ,c to ! con/ant factor on ,d & read ! sev]al products on ,d1 "u ! respective v>iable factors4 ,if ! factors are requir$ :i* w give a con/ant product 7really a case ( division71 set ! cursor to ! con/ant product on ,d4 ,!n obvi\sly1 z ! slide is mov$ along1 any pair ( factors f\nd simultane\sly "u ! cursor l9e on ,c1 & on ,d "u index ( ,c1 w give ! product4 ,a bett] me?od ( work+ w be expla9$ :5 we deal ) ! inv]sion ( ! slide4 ,x is sometimes useful to rememb] t al?\gh we usually set ! slide to ! rule1 we c obta9 ! result equally well by sett+ ! rule to ! slide4 ,?us1 br++ #a 7or #aj7 on ,d to #b on ,c1 we f9d on ,c1 ov] any o!r factor1 n on ,d1 ! product ( #b n4 ,b note t ! slide & rule h n[ *ang$ places1 & if we use rules = ! numb] ( digits 9 ! result1 we mu/ n[ d$uct #a f ! sum ( ! digits 9 ! factors1 :5 ! rule projects to ! "r ( ! slide4 ,) ! ord9>y #aj 94 rule x w be f\nd 9 g5]al t ! ext5t to :i* ! ,c & ,d scales are subdivid$ is su* z to enable n m ?an ?ree figures 9 ei!r factor be+ dealt )4 ,= ! same reason x is impossible to directly read m ?an ! fir/ ?ree figures ( any product1 al?\gh x is (t5 possible--by m5tally divid+ ! smalle/ space involv$ 9 ! read+--to correctly det]m9e ! f\r? figure ( a product4 ,necess>ily ?is me?od is only reliable :5 us$ 9 ! e>li] p>ts ( ! ,c & ,d scales4 ,h[ev]1 ! la/ num]al ( a ?ree-figure1 & 9 "s cases ! la/ ( a f\r-figure1 product c be readily asc]ta9$ by an inspection ( ! factors4 ,,ex4--#ai #bg #eac4 ,plac+ ! ,l4,h4 index ( ,c to #ai on ,d1 we f9d opposite #bg on ,c1 ! product1 :i* lies betwe5 #eaj & #eae4 ,a glance at ! factors1 h[ev]1 is suffici5t to decide t ! ?ird figure mu/ be #c1 s9ce ! product ( #i & #g is #fc1 & ! la/ figure ( ?is product mu/ be ! la/ figure 9 ! answ]4 ,,ex4--#gi #ia #gahi4 ,9 ?is case ! division l9e #ia on ,c indicates on ,d t ! answ] lies betwe5 #gahj & #gaij4 ,z ! la/ figure mu/ be #i1 x is at once inf]r$ t ! la/ two figures are #hi4 ,:5 !re are m ?an ?ree figures 9 ei!r or bo? ( ! factors1 ! f\r? & foll[+ figures to ! "r mu/ be neglect$4 ,x is well to note1 h[ev]1 t if ! fir/ neglect$ figure is #e1 or great] ?an #e1 x w g5]ally be advisable to increase by #a ! ?ird figure ( ! factor employ$4 ,g5]ally x w suffice to make ?is increase 9 "o ( ! two factors only1 b x is obvi\s t 9 "s cases great] accuracy w be obta9$ by increas+ bo? factors 9 ?is way4 ,,cont9u$ ,,multiplication4--,to f9d ! product ( m ?an two factors1 we make use ( ! cursor to m>k ! position ( successive products 7! value ( :i* does n conc]n u7 z ! sev]al factors are tak5 into ! calculation4 ,sett+ ! index ( ,c to ! #a/ factor on ,d1 we br+ ! l9e ( ! cursor to ! #bnd factor on ,c1 !n ! index ( ,c to ! cursor1 ! cursor to ! #crd factor1 index ( ,c to cursor1 & s on1 read+ ! f9al product on ,d "u ! la/ factor on ,c4 7,note t ! #a/ factor & ! result are read on ,d2 all int]m$iate read+s are tak5 on ,c47 ,if ! rule = ! numb] ( digits 9 a product is us$1 x is necess>y to note ! numb] ( times multiplication is effect$ ) ! slide project+ to ! "r4 ,?is numb]1 d$uct$ f ! sum ( ! digits ( ! sev]al factors1 gives ! numb] ( digits 9 ! product4 ,+5i\s devices h be5 adopt$ to record ! numb] ( times ! slide projects to ! "r1 b "s ( ^! are v inconv5i5t4 ,! au?or's me?od is to record ea* "t ! slide s projects1 by a m9us m>k1 ?us 4 ,^! c be not$ d[n 9 any conv5i5t mann]1 & ! sum ( ! m>ks s obta9$ d$uct$ f ! sum ( ! digits 9 ! sev]al factors1 gives ! numb] ( digits 9 ! product z be=e expla9$4 ,,ex4--#db #ga #ae #jcb #aba #agc1bjj4 ,! product giv51 :i* is t read on ! rule1 is obta9$ z foll[s3--,set ,r4,h4 index ( ,c to #db on ,d1 & br+ ! cursor to #ga on ,c4 ,next br+ ! ,l4,h4 index ( ,c to ! cursor1 & ! latt] to #ae on ,c4 ,?is multiplication is effect$ ) ! slide to ! "r1 & a memor&um ( ?is fact is kept by mak+ a m>k 4 ,br+ ! ,r4,h4 index ( ,c to ! cursor & ! latt] to #jcb on ,c4 ,!n set ! ,l4,h4 index ( ,c to ! cursor & read ! result1 #agcb1 on ,d "u #aba on ,c1 :ile z a slide aga9 projects to ! "r1 a second memo-m>k is record$4 ,!re are #b #b #a #j #c #h digits 9 ! factors1 & z !re 7 #b m>ks record$ dur+ ! op]ation1 !re w be #h #b #f digits 9 ! product1 :i* w !re=e read #agc1bjj 7#agc1aidef74 ,= a v r\gh evaluation ( ! result1 we note t #ae #jc is ab\t #je2 h5ce1 z a clue to ! numb] ( figures we h #dj #gj #fj #cjjj #fj #ahj1jjj4 ,,division4 ,! in/ructions = multiplication hav+ be5 giv5 9 "s detail1 a full discussion ( ! inv]se process ( division w be unnecess>y4 ,,rule ,,= ,,division4--,place ! divisor on ,c1 opposite ! divid5d on ,d1 & read ! quoti5t on ,d "u ! index ( ,c4 ,,ex4--#bbe #ah #abe4 ,br++ #ah on ,c to #bbe on ,d1 we f9d #abe "u ! ,l4,h4 index ( ,c4 ,z 9 multiplication1 ! factors are treat$ z :ole numb]s1 & ! position ( ! decimal po9t aft]w>ds decid$ accord+ to ! foll[+ rule1 :i*1 z w be se51 is ! rev]se ( t = multiplication3-- ,,rule ,,= ,,! ,,numb] ,,( ,,digits ,,9 ,a ,,quoti5t4--,if ! quoti5t is read ) ! slide project+ to ! ,,left1 subtract ! numb] ( digits 9 ! divisor f ^? 9 ! divid5d2 b if read ) ! slide to ! ,,"r1 ,,add #a to ?is diff]5ce4#b ,9 ! above example ! quoti5t is read (f ) ! slide to ! "r1 s ! numb] ( digits 9 ! answ] #c #b #a #b4 ,,ex4--#jjjjbba #jjag #jjac4 ,"h ! numb] ( digits 9 ! divid5d is #c1 & 9 ! divisor #a4 ,! diff]5ce is #b2 b z ! result is obta9$ ) ! slide to ! "r1 ?is result mu/ be increas$ by #a1 s t ! numb] ( digits 9 ! quoti5t is #b #a #a1 giv+ ! answ] z #jjac4 ,if pref]r$1 ! result c be obta9$ 9 ! mann] ref]r$ to :5 consid]+ ! multiplication ( decimals4 ,?us1 treat+ ! above z :ole numb]s1 we f9d t ! result ( divid+ #bba by #ag #ac1 s9ce ! diff]5ce 9 ! numb] ( digits 9 ! factors1 :i* is #a1 is1 ow+ to ! position ( ! slide1 increas$ by #a1 giv+ #b z ! numb] ( digits 9 ! answ]4 ,!n by ! rules = ! division ( decimals we "k t ! numb] ( decimal places 9 ! quoti5t is equal to #f #c #c1 %[+ t a cyph] is to be prefix$ to ! result read on ! rule4 ,z 9 multiplication1 s 9 division1 we h a ,,g5]al ,,rule ,,= ,,numb] ,,( ,,digits ,,9 ,a ,,quoti5t4--,:5 ! fir/ significant figure 9 ! ,,divisor is great] ?an t 9 ! ,,divid5d1 ! numb] ( digits 9 ! quoti5t is f\nd by subtract+ ! digits 9 ! divisor f ^? 9 ! divid5d4 ,:5 ! contr>y is ! case1 #a ,,is ,,to ,,be ,,add$ to ?is diff]5ce4 ,:5 ! fir/ figures are ! same1 ^? foll[+ mu/ be comp>$4 ,,e/imation ,,( ,,! ,,figures ,,9 ,a ,,quoti5t4--,! me?od ( r\ghly e/imat+ ! numb] ( figures 9 a quoti5t ne$s little explanation4 ,,ex4--#cie #ecdj #jjjjgd4 ,sett+ #ecd on ,c to #cie on ,d we read "u ! 7,r4,h47 index ( ,c1 ! significant figures on ,d1 :i* are #gd4 ,!n #ci #e is ab\t #jh & #jh #ajjj gives #jjjjh z a r\gh e/imate4 ,,ex4--#jjjjjjbhe #jjjjaig #jjaddf4 ,reg>d+ ?is z #bhe #ajf #aig #ajd we divide #bhe by #aig & obta9 #addf4 ,divid+ ! p[]s ( #aj we h #ajf #ajd #ajb1 s ! decimal po9t is to be mov$ two places to ! left & ! answ] is read z #jjaddf4 ,ano!r me?od ( divid+ des]ves m5tion z ( special s]vice :5 divid+ a numb] ( quantities by a con/ant divisor3--,set ! index ( ,c to ! divisor on ,d & ov] any divid5d on ,d1 read ! quoti5t on ,c4 ,= ! division ( a con/ant divid5d by a v>iable divisor1 set ! cursor to ! divid5d on ,d & br+ ! divisor on ,c successively to ! cursor1 read+ ! correspond+ quoti5ts on ,d "u ! index ( ,c4 ,ano!r me?od :i* avoids mov+ ! slide is expla9$ 9 ! section on 8,multiplication & ,division ) ! ,slide ,inv]t$40 ,,cont9u$ ,,division1 if we c s call su* an expression z 7#cad7_/7ghe #jjjjba #dc #fdd7 #jjfhh1 may be work$ by repeat+ z foll[s3--,set #ghe on ,c to #cad on ,d1 br+ cursor to index ( ,c1 #ba on ,c to cursor1 cursor to index1 #dc to cursor1 cursor to index1 #fdd to cursor1 & "u index ( ,c read #fhh on ,d z ! significant figures ( ! answ]4 ,= ! numb] ( figures 9 ! result1 we d$uct ! sum ( ! numb] ( digits 9 ! sev]al factors & add #a = ea* "t ! slide projects to ! "r1 :i* 9 ?is case occurs once4 ,!re are #c 7#c7 #a #b #c d5om9ator digits1 #a num]ator digit1 & #a is to be add$ to ! diff]5ce4 ,!re=e !re are #a #c #a #a digits 9 ! answ]1 :i* is !re=e #jjfhh4 ,! =ego+ me?od ( work+ may confuse ! beg9n]1 :o is apt to fall into ! process ( cont9u$ multiplication4 ,= ?is reason1 until famili>ity ) comb9$ me?ods has be5 acquir$1 ! product ( ! sev]al d5om9ators %\ld be fir/ f\nd by ! cont9u$ multiplication process1 & ! figures 9 ?is product det]m9$4 ,!n divide ! num]ator by ?is product to obta9 ! result4 ,z ! d5om9ator product w be read on ,d1 we may avoid resett+ ! slide by br++ ! num]ator on ,c to ?is product & read+ ! result on ,c ov] ! index ( ,d4 ,! slide & rule h "h *ang$ places2 h5ce if rules are foll[$ = ! numb] ( figures 9 ! result1 #a mu/ be add$ to ! diff]5ce ( digits1 :5 ! rule projects to ! "r ( ! slide4 ,! au?or's me?od ( record+ ! numb] ( times division is p]=m$ ) ! slide to ! "r is by v]tical memor&um m>ks1 ?us 4 ,! full significance ( ^! memo-m>ks w appe> 9 ! foll[+ section4 ,= a r\gh calculation to fix ! decimal po9t1 9 ?is example we move ! decimal po9ts 9 ! factors1 obta9+ 7#c7_/7jh #b #d #f7 7#c7_/7dj7 #jjge4 ,,! ,,use ,,( ,,! ,,upp] ,,scales ,,= ,,multiplication ,,& ,,division4 ,_m pref] to use ! upp] scales ,a & ,b1 9 pref]5ce to ,c & ,d4 ,! disadvantage is t z ! scales are only "o-half ! l5g? ( ,c or ,d1 ! graduation does n p]mit ( ! same degree ( accuracy be+ obta9$ z :5 work+ ) ! l[] scales4 ,b ! result c always be read directly f ! rule )\t ev] hav+ to *ange ! position ( ! slide aft] x has be5 initially set4 ,h5ce1 x obviates ! unc]ta9ty z to ! direction 9 :i* ! slide is to be mov$ 9 mak+ a sett+4 ,:5 ! ,a & ,b scales are employ$1 x is und]/ood t ! left-h& pair ( scales are to be us$ 9 ! same mann] z ,c & ,d1 & s f> ! rules relat+ to ! latt] are entirely applicable4 ,b 9 ?is case ! slide is always mov$ to ! "r1 s t 9 multiplication ! product is f\nd ei!r ^u ! left or "r scales ( ,a4 ,if x is f\nd on ! left ,a scale1 ! rule = ! numb] ( digits 9 ! product is f\nd z = ! ,c & ,d scales1 & is equal to ! sum ( ! digits 9 ! two factors1 m9us #a2 b if f\nd on ! "r-h& ,a scale1 ! numb] ( digits 9 ! product is equal to ! sum ( ! digits 9 ! two factors4 ,9 division1 simil> modifications are necess>y4 ,if :5 mov+ ! slide to ! "r ! division c be completely effect$ by us+ ! ,l4,h4 scale ( ,a1 ! quoti5t 7read on ,a above ! ,l4,h4 ( index ,b7 has a numb] ( digits equal to ! numb] 9 ! divid5d1 less ! numb] 9 ! divisor1 plus #a4 ,b if ! division necessitates ! use ( bo? ! ,a scales1 ! numb] ( digits 9 ! quoti5t equals ! numb] 9 ! divid5d1 less ! numb] 9 ! divisor4 ,,reciprocals4 ,a special case ( division to be consid]$ is ! det]m9ation ( ! reciprocal ( a numb] n1 or 7#a7_/7n74 ,foll[+ ! ord9>y rule = division1 x is evid5t t sett+ n on ,c to #a on ,d1 gives 7#a7_/7n7 on ,d "u #a on ,c4 ,x is m important to obs]ve t by inv]t+ ! op]ation--sett+ #a 7or #aj7 on ,c to n on ,d--we c read 7#a7_/7n7 on ,c ov] #a 7or #aj7 on ,d4 ,h5ce :5ev] a result is read on ,d "u an index ( ,c1 we c also read its reciprocal on ,c ov] :i*ev] index ( ,d is available4 ,! ,numb] ( ,digits 9 a ,reciprocal is obvi\s :5 n #aj1 #ajj1 or any p[] 7p7 ( #aj4 ,?us 7#a7_/7aj7 #ja2 7#a7_/7ajj7 #jja2 7#a7_/7ajp7 #a prec$$ by p #a cyph]s4 ,= all o!r cases we h ! rule3--,subtract f #a ! numb] ( digits 9 ! numb]4 ,,ex4--7#a7_/7cci7 #jjjbie4 ,!re are #c digits 9 ! numb]2 h5ce1 !re are #a #c #b digits 9 ! answ]4 ,,ex4--7#a7_/7jjjjjaef7 #fd1ajj4 ,!re are #d digits 9 ! numb]2 h5ce1 !re are #a 7#d7 #e digits 9 ! result4 ,,cont9u$ ,,multiplication ,,& ,,division4 ,by comb9+ ! rules = multiplication & division1 we c readily evaluate expressions ( ! =m 7a7_/7b7 7c7_/7d7 7e7_/7f7 7g7_/7h7 x4 ,! simple/ case1 7a c7_/7b7 c be solv$ by "o sett+ ( ! slide4#c ,take z an example1 7#adde #fj7_/7he7 #ajb4 ,sett+ #he on ,c to #adde on ,d1 we c1 if desir$1 read #ag on ,d "u #a on ,c1 z ! quoti5t4 ,h[ev]1 we are n conc]n$ ) ?is1 b require its multiplication by #fj1 & ! slide be+ already set = ?is op]ation1 we at once read "u #fj on ,c ! result1 #ajb1 on ,d4 ,! figures 9 ! answ] are obvi\s4 ,:5 !re are m factors to take into acc\nt1 we place ! cursor ov] #ajb on ,d1 br+ ! next divisor on ,c to ! cursor1 move ! cursor to ! next multipli] on ,c1 br+ ! next divisor on ,c to ! cursor1 & s on1 until all ! factors h be5 dealt )4 ,note t only ! fir/ factor & ! result are read on ,d2 also t ! cursor is mov$ = multiply+ & ! slide = divid+4 ,numb] ( ,digits 9 ,result 9 ,comb9$ ,multiplication & ,division4--,= ^? :o use rules ! au?or's me?od ( det]m9+ ! decimal po9t 9 comb9$ multiplication & division may be us$4 ,ea* "t multiplication is p]=m$ ) ! slide project+ to ! "r1 make a m>k2 ea* "t division is effect$ ) ! slide to ! "r1 make a m>k2 b all[ ! m>ks to cancel ! m>ks z f> z !y w4 ,subtract ! sum ( ! digits 9 ! d5om9ator f ! sum ( digits 9 ! num]ator1 & to ?is diff]5ce add any uncancell$ memo-m>ks1 if ( "*1 or subtract !m if ( "*4 ,,ex4--7#dce #bid #ea #cb7_/7bg #chc #aje #aca7 #adfh4 ,sid5ote3 ,set #bg on ,c to #dce on ,d1 & z ) ?is division ! slide is to ! "r1 make ! fir/ m>k4 ,br+ cursor to #bid on ,c1 & z 9 ?is multiplication ! slide is to ! "r1 make ! fir/ m>k1 cancell+ z %[n4 ,sett+ #chc on ,c to ! cursor1 requires ! second m>k1 :i*1 h[ev]1 is cancell$ 9 turn by ! multiplication by #ea4 ,! division by #aje requires ! ?ird m>k1 & aft] multiply+ by #cb 7requir+ no m>k7 ! f9al division by #aca requires ! f\r? m>k4 ,!n1 z !re are #h num]ator digits1 #f d5om9ator1 & #b uncancell$ memo-m>ks 7:i*1 be+ #a1 are additive7 we h ,numb] ( digits 9 result #h #f #b #d4 ,_h ! uncancell$ m>ks be5 9 "*1 ! numb] ( digits w\ld h be5 #h #f #b #j4 ,= quantities less ?an #ja ! digit place numb]s w be negative4 ,! tr\blesome addition ( ^! may be avoid$ by transf]r+ !m to ! opposite side & treat+ !m z positive4 #b #d #jjjcef #bga #jjhcge ,?us3-- #bhh #jadbf #ihe #jjjjjb #b #a #a ,! fir/ num]ator1 #jjjcef1 has #b digits4 ,note ?is by plac+ #b bel[ ! l[] l9e z %[n4 #bga has #b digits2 place #b ov] x4 #jjhcge has #a digit2 h5ce place #a bel[ ! l[] l9e4 ,! fir/ d5om9ator has no digits2 ! second1 #ihe1 has #a digit2 h5ce place #a "u x4 #jjjjjb has #d digits2 place #d above ! upp] l9e4 ,! sum ( ! top s]ies is #b #d #f2 ( ! bottom s]ies #b #a #a #d4 ,subtract+ ! bottom f ! top1 we h #f #d #b digits1 to :i* #a has to be add$ = an uncancell$ memo-m>k1 & ! result is read z #bhh4 ,mov+ ! decimal po9t (t5 facilitates matt]s4 ,?us1 7#cbd #jih #dcb #jjbag7_/7dga #jage #jjjjjjfba #dabjjj7 is mu* m conv5i5tly dealt ) :5 re-arrang$ z 7#cbd #ih #dcb #bag7_/7dga #age #fba #dab7 #ada4 ,to det]m9e ! numb] ( figures 9 ! result by r\gh cancell+ & m5tal calculation1 we note t #dga ent]s #dcb ab\t #ajj times2 #ih ent]s #age ab\t #b2 #fba into #cbd ab\t #e2 & #bag into #dab ab\t #b4 ,?is gives 7#ejj7_/7d7 #abe1 %[+ t ! result conta9s #c digits4 ,f ! slide rule we read #ada1 :i* is !re=e ! result s\ght4 ,! occasional trav]s+ ( ! slide ?r\gh ! rule1 to int]*ange ! indices--a cont+5cy :i* ! use ( ! ,c & ,d scales always involves--may (t5 be avoid$ by a v simple exp$i5t4 ,su* an example z 7#fai #cab #dbb7_/7aabj #hhf #b4ji7 #cic is sometimes cit$ z a p>ticul>ly difficult case4 ,work+ ?r\gh ! expression z giv51 two trav]s+s ( ! slide are necess>y2 b by tak+ ! factors 9 ! slightly diff]5t ord]1 7#fai #cab #dbb7_/7hhf #bji #aabj71 s t ! significant figures ( ea* pair are m ne>ly alike1 we n only avoid any trav]s+ ! slide1 b we also r$uce ! ext5t to :i* ! slide is mov$ to effect ! sev]al divisions4 ,su* cases z 7a b7_/7c d e f g7 or 7a b c d e7_/7f g7 really resolve !mselves into 7a b #a #a #a7_/7;c d e f g7 & 7a b c d e7_/7f g #a #a #a71 b1 ( c\rse1 if rules are us$ to locate ! decimal po9t1 ! #a's s 7m5tally7 introduc$ are n to be c\nt$ z additional figures 9 ! factors4 ,,multiplication ,,& ,,division ,,) ,,! ,,slide ,,inv]t$4 ,if ! slide be inv]t$ 9 ! rule b ) ! same face upp]mo/1 s t ! scale lies adjac5t to ! ,a scale1 & ! "r & left indices ( ! slide & rule are plac$ 9 co9cid5ce1 we f9d ! product ( any numb] on ,d by ! co9cid5t numb] on 7readily ref]r$ to ea* o!r by ! cursor7 is always #aj4 ,h5ce1 by read+ ! numb]s on z decimals1 we h ov] any unit numb] on ,d1 its reciprocal on 4 ,?us #b on ,d is f\nd opposite #je on 2 #c on ,d opposite to #jccc2 :ile opposite #h on is #jabe on ,d1 etc4 ,! reason ( ?is is t ! sum ( ! l5g?s ( ! slide & rule correspond+ to ! factors1 is always equal to ! l5g? correspond+ to ! product--9 ?is case1 #aj4 ,x w be se5 t if we attempt to apply ! ord9>y rule = multiplication1 ) ! slide inv]t$1 we %all actually be multiply+ ! "o factor tak5 on ,d by ! reciprocal ( ! o!r tak5 on 4 ,b multiply+ by ! reciprocal ( a numb] is equival5t to divid+ by t numb]1 & divid+ a factor by ! reciprocal ( a numb] is equival5t to multiply+ by t numb]4 ,x foll[s t ) ! slide inv]t$ ! op]ations ( multiplication & division are rev]s$1 z are also ! rules = ! numb] ( digits 9 ! product & ! position ( ! decimal po9t4 ,h5ce1 9 multiply+ ) ! slide inv]t$1 we place 7by ! aid ( ! cursor7 "o factor on opposite ! o!r factor on ,d1 & read ! result on ,d "u ei!r index ( 4 ,x foll[s t ) ! slide ?us set1 any pair ( co9cid+ factors on & ,d w give ! same con/ant product f\nd on ,d "u ! index ( 4 ,"o useful application ( ?is fact is f\nd 9 select+ ! scantl+s ( rectangul> sections ( giv5 areas or 9 decid+ ^u ! dim5sions ( rectangul> %eets1 plates1 ci/]ns1 etc4 ,?us by plac+ ! index ( to #gb on ,d1 x is readily se5 t a plate hav+ an area ( #gb sq4 ft4 may h sides #h by #i ft41 #f by #ab1 #e by #add1 #d by #ah1 #c by #bd1 #b by #cf1 ) innum]able int]m$iate values4 ,_m o!r useful applications ( a simil> "* w sugge/ !mselves4 ,,proportion4 ,) ! slide 9 ! ord9>y position & ) ! indices ( ! ,c & ,d scales 9 exact agreem5t1 ! ratio ( ! correspond+ divisions ( ^! scales is #a4 ,if ! slide is mov$ s t #a on ,c agrees ) #b on ,d1 we "k t "u any numb] n on ,c is n #b on ,d1 s t if we read num]ators on ,c & d5om9ators on ,d we h ,c #a #ae #b #c #d ,d#a #b #c #d #f #h4 ,9 o!r words1 ! numb]s on ,d be> to ! co9cid+ numb]s on ,c a ratio ( #b to #a4 ,obvi\sly ! same condition w obta9 no matt] 9 :at position ! slide may be plac$4 ,! rule = proportion1 :i* is app>5t f ! =ego+1 may be express$ z foll[s3-- ,,rule ,,= ,,proportion4--,set ! fir/ t]m ( a proportion on ! ,c scale to ! second t]m on ! ,d scale1 & opposite ! ?ird t]m on ! ,c scale read ! f\r? t]m on ! ,d scale4 ,,ex4--,f9d ! #d? t]m 9 ! proportion ( #bj #bg #gj x4 ,set #bj on ,c to #bg on ,d1 & opposite #gj on ,c read #ide on ,d4 ,?us ,c #bj #gj ,d #bg #ide4 ,x w be evid5t t ?is is m]ely a case ( comb9$ multiplication & division ( ! =m1 7#bj #gj7_/7bg7 #ide4 ,h5ce1 giv5 any ?ree t]ms ( a proportion1 we set ! #a/ to ! #bnd1 or ! #crd to ! #d?1 z ! case may be1 & opposite ! o!r giv5 t]m read ! t]m requir$4#d ,?us1 9 r$uc+ vulg> fractions to decimals1 ! decimal equival5t ( 7#c7_/7af7 is det]m9$ by plac+ #c on ,c to #af on ,d1 :5 ov] ! index or #a ( ,d we read #jahge on ,c4 ,9 ?is case ! t]ms are #c #af x #a4 ,= ! inv]se op]ation--to f9d a vulg> fraction equival5t to a giv5 decimal--! giv5 decimal fraction on ,c is set to ! index ( ,d1 & !n opposite any d5om9ator on ,d is ! correspond+ num]ator ( ! fraction on ,c4 ,if ! index ( ,c be plac$ to agree ) #cadaf on ,d1 x w be cle> f :at has be5 said t ?is ratio exi/s ?r\gh\t betwe5 ! numb]s ( ! two scales4 ,!re=e1 aga9/ any diamet] ( a circle on ,c w be f\nd ! correspond+ circumf]5ce on ,d4 ,9 ! same way1 by sett+ #a on ,c to ! appropriate conv]sion factor on ,d1 we c conv]t a s]ies ( values 9 "o d5om9ation to _! equival5ts 9 ano!r d5om9ation4 ,9 ?is connection ! foll[+ table ( conv]sion factors w be f\nd ( s]vice4 ,if ! ,a & ,b scales are us$ in/ead ( ! ,c & ,d scales1 a complete set ( conv]sions w be at once obta9$4 ,9 ?is case1 h[ev]1 ! left-h& ,a & ,b scales %\ld be us$ = ! initial sett+1 any values read on ! "r-h& ,a or ,b scales be+ read z ( t5fold value4 ,) ! ,c & ,d scales a portion ( ! "o scale w project beyond ! o!r4 ,to read ?is portion ( ! scale1 ! cursor or runn] is br\ght to :i*ev] index ( ! ,c scale falls )9 ! rule1 & ! slide mov$ until ! o!r index ( ! ,c scale co9cides ) ! cursor1 :5 ! rema9d] ( ! equival5t values c !n be read (f4 ,x mu/ be rememb]$ t if ! slide is mov$ 9 ! direction ( notation 7to ! "r71 ! values read !reon h a t5fold great] value2 if ! slide is mov$ to ! left1 ! read+s !reon are decreas$ 9 a t5fold degree4 ,al?\gh pref]r$ by _m1 9 ! =m giv51 ! case is obvi\sly "o ( multiplication1 & is s treat$ 9 ! ,data ,slips at ! end ( ! book4 ,,table ,,( ,,conv]sion ,,factors4 ,,geometrical ,,equival5ts4 ,,scale ,c4 ,,scale ,d4 ,if ,c #a1 ,d ,diamet] ( circle ,circumf]5ce ( circle #cadaf ,side ( inscrib$ squ>e #jgjg equal squ>e #jhhf equilat]al triangle #acdf ,circum4 ( circle inscrib$ squ>e #jbbe equal squ>e #jbhb ,side ( squ>e ,diagonal ( squ>e #adad ,squ>e in* ,circul> in* #abgc ,area ( circle ,area ( inscrib$ squ>e #jfcf ,,measures ,,( ,,l5g?4 ,in*es ,millimetres #bedj ,c5timetres #bed #h?s ( an in* ,millimetres #cage #af?s #aehg #cbnds #jgid #fd?s #jcig ,feet ,metres #jcjdh ,y>ds #jiadd ,*a9s #bjaaf ,miles ,kilometres #afji ,,measures ,,( ,,area4 ,squ>e in*es ,squ>e c5timetres #fdf ,circul> #ejfg ,squ>e feet metres #jjibi y>ds #jhcf miles kilometres #bei ,hect>es #beijj ,acres #jdjdf ,,measures ,,( ,,capacity4 ,cubic in*es ,cubic c5timetres #afch ,imp]ial gallons #jjjcfj ,u4,s4 gallons #jjjdcb ,litres #jjafch ,cubic feet ,cubic metres #jjbhc ,imp]ial gallons #fbc ,u4,s4 gallons #gdh ,litres #bhcg y>ds ,cubic metres #jgfd ,imp]ial gallons ,litres #ded ,u4,s4 gallons #abjj ,bu%els ,cubic metres #jjcfc feet #abhc ,,measures ,,( ,,weight4 ,gra9s ,grammes #jjfdh ,ounces 7,troy7 #caajc 7,avoird47 #bhce ,kilogrammes #jjbhce ,p\nds 7,troy7 #jcgcb 7,avoird47 #jdecf ,hundr$weights #ejhjb ,tons #ajafd ,metric tonnes #ajaf ,,comp\nd ,,factors--,,velocities4 ,feet p] second ,metres p] second #jcjdh m9ute #ahbhh ,miles p] h\r #j4fhb m9ute ,met]s p] second #jjjejh m9ute #jcjdh ,miles p] h\r #jjaacf ,y>ds p] #jjcda ,miles p] h\r ,metres p] m9ute #bfhb ,knots #cjhh ,miles p] h\r #aaea ,,comp\nd ,,factors--,,pressures4 ,p\nds p] sq4 in* ,grammes p] sq4 mm4 #jgjca ,kilos4 p] sq4 c5timetre #jjgjc ,atmosph]es #jjfh ,head ( wat] 9 in*es #bgga feet #bcji metres #jgeg ,in*es ( ,m]cury #bjd ,in*es ( wat] ,p\nds p] squ>e in* #jjcfa ,in*es ( m]cury #jjgad ,p\nds p] squ>e foot #ebj ,in*es ( m]cury ,atmosph]es #jjccc ,atmosph]es ,metres ( wat] #ajcd ,kilos4 p] sq4 cm4 #ajcc ,feet ( wat] ,p\nds p] squ>e foot #fbce ,atmosph]es #jjbid ,in*es ( m]cury #jhhc ,p\nds p] sq4 foot #jjadag ,kilos4 p] squ>e metre #dhhc ,atmosph]es #jjjjdgb ,p\nds p] sq4 y>d ,kilos4 p] squ>e metre #jedbe ,tons p] sq4 in* squ>e mm4 #aege sq4 foot ,tonnes p] squ>e metre #ajicf ,,comp\nd ,,factors--,,weights1 ,,capacities1 ,,etc4 ,p\nds p] l9eal ft4 ,kilos4 p] l9eal metre #adhh p] l9eal yd4 #jdif p] l9eal mile ,kilos4 p] kilometre #jbhah ,tons ,tonnes #jfcac ,feet ,metres #ahid ,p\nds p] cubic 94 ,grammes p] cubic cm4 #bgfh p] cubic ft4 ,kilos4 p] cubic metre #afjb p] cubic yd4 #jeic ,tons p] cubic y>d ,tonnes #acbi ,cubic yds4 p] p\nd ,cubic metres p] kilo4 #afhe p] ton p] tonne #jgebe ,cubic in* ( wat] ,weight 9 p\nds #jjcfjh ,cubic feet ( wat] #fbce kilos #bhbc ,imp]ial gallons #fbce ,u4,s4 gallons #gdh ,litre ( wat] ,cubic in*es #fajbe ,gallons ( wat] ,weight 9 kilos #ded ,p\nds ( fre% wat] ,p\nds ( sea wat] #ajbf ,gra9s p] gallon ,grammes p] litre #jjadbf ,p\nds p] gallon ,kilos4 p] litre #jjiih p] ,u4,s4 gal4 #jaae ,,comp\nd ,,factors--,,p[] ,,units1 ,,etc4 ,briti% ,!r4 ,units4 ,kilogrammetres4 #ajh ,j\les #ajeh ,calories 7,fr4 ,!r4 units7#jbeb p] sq4 ft4 p] squ>e metre #bgac p] p\nd p] kilogramme #jeee ,p\nds p] sq4 ft4 ,dynes1 p] sq4 cm4 #dgi ,foot-p\nds ,kilogrammetres #jachb ,j\les #acef ,!rmal ,units #jjjabi ,calorie #jjjjcbd ,foot-tons ,tonne-metres #jccc ,horse-p[] ,=ce de*eval 7,fr4,h4,p47 #ajad ,kil[atts #jgdf ,p\nds p] ,h4,p4 ,kilos4 p] *eval #jddg ,squ>e feet p] ,h4 ,p4 ,squ>e metres p] *eval #jjaif ,cubic ,cubic #jjbgi ,watts ,!r4 ,units p] h\r #cdd ,foot-p\nds p] second #jgc p] m9ute #ddbd ,watt-h\rs ,kilogrammetres #cfg ,j\les #cfjj ,kilogrammetres #ihjf ,inv]se ,proportion4--,if 8m0 requires 8less10 or 8less0 requires 8m10 ! case is "o ( inv]se proportion1 & al?\gh x w be se5 t ?is =m ( proportion is q readily dealt ) by ! prec$+ me?od1 ! work+ is simplifi$ to "s ext5t by inv]t+ ! slide s t ! ,c scale is adjac5t to ! ,a scale4 ,by ! aid ( ! cursor1 ! values on ! inv]t$ ,c 7or 7 scale1 & on ! ,d scale1 c be !n read (f4 ,^! w n[ con/itute a s]ies ( inv]se ratios4 ,= example1 9 ! proportion #h #d ,d #ae #c ! #d on ! scale is br\ght opposite #c on ,d1 :5 "u #h on is f\nd #ae on ,d4#e ,,g5]al ,,h9ts ,,on ,,! ,,elem5t>y ,,uses ,,( ,,! ,,slide ,,rule4 ,be=e ! m complex op]ations ( involution1 evolution1 etc41 are consid]$1 a few g5]al h9ts on ! use ( ! slide rule = elem5t>y op]ations may be ( s]vice1 especially z ^! w s]ve to en=ce "s ( ! m important po9ts br\ght out 9 ! prec$+ sections4 ,always use ! slide rule 9 z direct a light z possible4 ,/udy ! mann] 9 :i* ! scales are divid$4 ,foll[ ! graduations ( ! ,c & ,d scales f #a to #aj1 not+ ! values giv5 by ea* successive graduation & h[ ^! values *ange z we foll[ along to ! "r4 ,d ! same ) ! two halves ( ! ,a & ,b scales & note ! diff]5ce 9 ! value ( ! subdivisions1 due to ! %ort] scale-l5g?s4 ,practise read+ values by sett+ #a on ,c to "s value on ,d & read+ "u #b1 #c1 #d1 etc41 on ,c1 *eck+ ! read+s by m5tal ari?metic4 ,to ! same end1 f9d squ>es1 squ>e roots1 etc41 comp>+ ! results ) ! actual values z giv5 9 tables4 ,practise sett+ bo? slide & cursor to values tak5 at r&om4 ,aim at accuracy2 spe$ w come ) practice4 ,:5 9 d\bt z to any me?od ( work+1 v]ify by mak+ a simple calculation ( ! same =m4 ,foll[ ! or?odox me?ods ( work+ until entirely confid5t 9 ! use ( ! in/rum5t1 & ev5 !n d n readily make a *ange4 ,if any alt]$ proc$ure is adopt$1 fir/ "w a simple case & gu>d c>efully aga9/ unconsci\sly laps+ into ! usual me?od dur+ ! op]ation4 ,unless ! calculation is ( a /raight=w>d "*1 "t tak5 9 consid]+ h[ be/ to attack x 7re>rang+ ! expression if desirable7 is g5]ally "t well sp5t4 ,9 sett+ two values toge!r1 set ! cursor to "o ( !m on ! rule1 & br+ ! o!r1 on ! slide1 to ! cursor l9e4 ,9 multiply+ factors1 z #eg #jabef1 take ! fractional value fir/4 ,x is easi] to set #a on ,c to #abef on ,d & read "u #eg on ,c1 ?an to rev]se ! proc$ure4 ,:5 bo? values are eye-e/imat$1 set ! cursor to ! second factor on ,c & read ! result on ,d1 "u ! cursor l9e4 ,9 cont9u\s op]ations avoid mov+ ! slide fur!r ?an necess>y1 by tak+ ! factors 9 t ord] :i* w keep ! scale read+s z close toge!r z possible4 ,,squ>es ,,& ,,squ>e ,,roots4 ,we h se5 t ! relation :i* ! upp] scales be> to ! l[] set is su* t ov] any numb] on ,d is its squ>e on ,a1 &1 conv]sely1 "u any numb] on ,a is its squ>e root on ,d1 ! same rem>ks apply+ to ! ,c & ,b scales on ! slide4 ,tak+ ! values engrav$ on ! rule1 we h on ,d1 numb]s ly+ betwe5 #a & #aj1 & on ,a ! correspond+ squ>es ext5d+ f #a to #ajj4 ,h5ce ! squ>es ( numb]s betwe5 #a & #aj1 or ! roots ( numb]s betwe5 #a & #ajj1 c be read (f on ! rule by ! aid ( ! cursor4 ,all o!r cases are br\ght )9 ^! ranges ( values by factoris+ ) p[]s ( #aj1 z be=e expla9$4 ,! m practical rule is ! foll[+3-- ,to ,f9d ! ,squ>e ( a ,numb]1 set ! cursor to ! numb] on ,d & read ! requir$ squ>e on ,a "u ! cursor4 ,! rule = ,! ,numb] ( ,digits 9 a ,squ>e is easily d$ucible f ! rule = multiplication4 ,if ! squ>e is read on ! left scale ( ,a1 x w conta9 twice ! numb] ( digits 9 ! orig9al numb] less #a2 if x is read on ! "r scale ( ,a1 x w conta9 twice ! numb] ( digits 9 ! orig9al numb]4 ,,ex4--,f9d ! squ>e ( #aad4 ,plac+ ! cursor to #aad on ,d1 x is se5 t ! co9cid+ numb] on ,a is #ac4 ,z ! result is read (f on ! left scale ( ,a1 ! numb] ( digits w be 7#c #b7 #a #e1 & ! answ] is read z #ac1jjj4 ,! true result is #ab1iif4 ,,ex4--,f9d ! squ>e ( #jjjic4 ,! cursor be+ plac$ to #ic on ,d1 ! numb] on ,a is f\nd to be #hfe4 ,! result is read on ! "r scale ( ,a1 s ! numb] ( digits #b #b #d1 & ! answ] is read z #jjjjjhfe #jjjjjhfdi4 ,squ>e ,root4--,! =ego+ rules sugge/ ! me?od ( proc$ure 9 ! inv]se op]ation ( extract+ ! squ>e root ( a giv5 numb]1 :i* w be f\nd on ! ,d scale opposite ! numb] on ! ,a scale4 ,x is necess>y to obs]ve1 h[ev]1 t if ! numb] consi/s ( an odd numb] ( digits1 x is to be tak5 on ! left-h& portion ( ! ,a scale1 & ! numb] ( digits 9 ! root 7,n #a7_/7b71 ,n be+ ! numb] ( digits 9 ! orig9al numb]4 ,:5 !re is an ev5 numb] ( digits 9 ! numb]1 x is to be tak5 on ! "r-h& portion ( ! ,a scale1 & ! root conta9s "o-half ! numb] ( digits 9 ! orig9al numb]4 ,,ex4--,f9d ! squ>e root ( #cf1ejj4 ,z !re is an odd numb] ( digits1 plac+ ! cursor to #cfe on ! ,l4,h4 ,a scale gives #aia on ,d4 ,by ! rule !re are 7,n #a7_/7b7 7#e #a7_/7b7 #c digits 9 ! requir$ root1 :i* is !re=e read z #aia #aiaje4 ,,ex4--,f9d 7#jjjih47 ,plac+ ! cursor to #ih on ! "r-h& scale ( ,a 7s9ce #b is an ev5 numb] ( digits71 x is se5 t ! co9cid+ numb] on ,d is #ii4 ,z ! numb] ( digits 9 ! numb] is #b1 ! numb] ( digits 9 ! root w be 7#b7_/7b7 #a4 ,x w !re=e be read z #jjii #jjihii4 ,,ex4--,f9d 7#jjih74 ,! numb] ( digits is #a1 s "u #ih on ! left scale ( ,a1 we f9d #cac on ,d4 ,by ! rule ! numb] 9 ! root w be 7#a #a7_/7b7 #j1 & ! root is !re=e read z #jcac #jcacjdi4 ,,ex4--,f9d 7#jadi47 ,z ! numb] ( digits 7#j7 is ev51 ! cursor is set to #adi on ! "r-h& scale ( ,a1 giv+ #chf on ,d4 ,by ! rule1 ! numb] ( digits 9 ! root w be 7#j7_/7b7 #j1 & ! root w be read z #jchf #jchfje4 ,ano!r me?od ( extract+ ! squ>e root1 by :i* m accurate read+s may g5]ally be obta9$1 is by us+ ! ,c & ,d scales only1 ) ! slide inv]t$4 ,if !re is an odd numb] ( digits 9 ! numb]1 ! "r index1 or if an ev5 numb] ( digits ! left index1 ( ! inv]t$ scale is plac$ s z to co9cide ) ! numb] on ,d ( :i* ! root is s\ght4 ,!n ) ! cursor1 ! numb] is f\nd on ,d :i* co9cides ) ! same numb] on 1 :i* numb] is ! root s\ght4 ,,ex4--,f9d 7#bbb47 ,plac+ ! left index ( to #bbb on ,d1 ! two equal co9cid+ numb]s on & ,d are f\nd to be #dga4 ,note t "u ! cursor l9e we h ! orig9al numb]1 #bbb1 on ,a1 & f ?is ! numb] ( digits 9 ! root is det]m9$ z be=e4 ,! plan ( f9d+ ! squ>e ( a numb] by ord9>y multiplication is (t5 v conv5i5t4 ,! inv]se process ( f9d+ a squ>e root by trial division is n to be recomm5d$4 ,to obta9 a close value ( a root or to v]ify "o f\nd 9 ! usual way1 ! au?or has1 on occasion1 adopt$ ! foll[+ plan3--,set #a 7or #aj7 on ,b to ! numb] on ! ,a scale 7,l4,h4 or ,r4,h4 z ! case may require71 & br+ ! cursor to ! numb] on ,d4 ,if ! root f\nd is correct1 ! read+s on ,c "u ! cursor & on ,d "u ! index ( ,c1 w be 9 exact agreem5t4 ,if #a on ,b is plac$ to a numb] n on ! ,l4,h4 ,a scale1 ! /ud5t w note t :ile root n is read on ,d "u #a on ,c1 ! root ( #aj n is read on ,d "u #aj on ,b4 ,h5ce1 if pref]r$1 ! numb] c be tak5 always on ! fir/ scale ( ,a & ! root read "u #a or #aj on ,b1 accord+ to :e!r !re is an odd or ev5 numb] ( digits 9 ! numb]4 ,obvi\sly ! second root is ! fir/ multipli$ by 7#aj74 ,,cubes ,,& ,,cube ,,roots4 ,9 rais+ a numb] to ! ?ird p[]1 a comb9ation ( ! prec$+ me?od & ord9>y multiplication is employ$4 ,,to ,,f9d ,,! ,,cube ,,( ,a ,,numb]4--,set ! ,l4,h4 or ,r4,h4 index ( ,c to ! numb] on ,d1 & opposite ! numb] ,,on ,,! ,,left-,,h& scale ( ,b read ! cube on ! ,l4,h4 or ,r4,h4 scale ( ,a4 ,by ?is rule f\r scales are br\ght into requisition4 ,( ^!1 ! ,d scale & ! ,l4,h4 ,b scale are always employ$1 & are to be read z ( equal d5om9ation4 ,! values assign$ to ! ,l4,h4 & ,r4,h4 scales ( ,a w be app>5t f ! foll[+ consid]ations4 ,comm5c+ ) ! indices ( ,c & ,d co9cid+1 & mov+ ! slide to ! "r1 x w be se5 t1 work+ 9 accordance ) ! above rule1 ! cubes ( numb]s f #a to #baed 7 7#aj77 w be f\nd on ! fir/ or ,l4,h4 scale ( ,a4 ,mov+ ! slide /ill f>!r to ! "r1 we obta9 on ! ,r4,h4 ,a scale cubes ( numb]s f #baed to #dfda 7or 7#aj7 to 7#ajj774 ,_h we a ?ird repetition ( ! ,l4,h4 ,a scale1 ! ,l4,h4 index ( ,c c\ld be /ill fur!r trav]s$ to ! "r1 & ! cubes ( numb]s f #dfda to #aj read (f on ?is prolongation ( ,a4 ,b ! same end c be atta9$ by mak+ use ( ! ,r4,h4 index ( ,c1 :51 trav]s+ ! slide to ! "r z be=e1 ! cubes ( numb]s f #dfda to #aj on ,d c be read (f on ! ,l4,h4 ,a scale ov] ! correspond+ numb]s on ! ,l4,h4 ,b scale4 ,h5ce1 us+ ! ,l4,h4 index ( ,c1 ! read+s on ! ,l4,h4 ,a scale may be reg>d$ comp>atively z units1 ^? on ! ,r4,h4 ,a scale z t5s2 :ile = ! hundr$s we aga9 make use ( ! ,l4,h4 ,a scale 9 conjunction ) ! "r-h& index ( ,c4 ,by keep+ ^! po9ts 9 view1 ! numb] ( digits 9 ! cube 7,n7 ( a giv5 numb] 7n7 are readily d$uc$4 ,?us1 if ! units scale is us$1 ,n #cn #b2 if ! t5s scale1 ,n #cn #a2 :ile if ! hundr$s scale be us$1 ,n #cn4 ,plac$ 9 ! =m ( rules3-- ,n #cn #b :5 ! product is read on ! ,l4,h4 scale ( ,a ) ! slide to ! "r 7units scale74 ,n #cn #a :5 ! product is read on ! ,r4,h4 scale ( ,a2 slide to ! "r 7t5s scale74 ,n #cn :5 ! product is read on ! ,l4,h4 scale ( ,a ) ! slide to ! left 7hundr$s scale74 ,) decimals ! same rule applies1 b1 z be=e1 ! numb] ( digits mu/ be read z #a1 #b1 etc41 :5 "o1 two1 etc41 cyph]s foll[ imm$iately aft] ! decimal po9t4 ,,ex4--,f9d ! value ( #adc4 ,plac+ ! ,l4,h4 index ( ,c to #ad on ,d1 ! read+ on ,a opposite #ad on ! ,l4,h4 scale ( ,b is f\nd to be ab\t #bgde #bgdd4 ,,ex4--,f9d ! value ( #bfdc4 ,plac+ ! ,l4,h4 index ( ,c to #bfd on ,d1 ! read+ on ,a opposite #bfd on ! ,l4,h4 scale ( ,b is f\nd to be ab\t #ah1djj #ah1ciigdd4 ,,ex4--,f9d ! value ( #gcc4 ,9 ?is case x becomes necess>y to use ! ,r4,h4 index ( ,c1 :i* is set to #gc on ,d1 :5 opposite #gc on ! ,l4,h4 scale ( ,b is read #chi #chijag on ,a4 ,,ex4--,f9d ! value ( #jjgcc4 ,f ! sett+ z be=e x is se5 t ! numb] ( digits 9 ! numb] mu/ be multipli$ by #c4 ,h5ce1 z !re is #a digit 9 #jjgc1 !re w be #c 9 ! cube1 :i* is !re=e read #jjjjchi4 ,! la/ two examples s]ve to illu/rate ! pr9ciple ( factoris+ ) p[]s ( #aj4 ,?us #jjgc #gc #ajb2 #jjgcc #gcc 7#ajb7c #chi #ajf #jjjjchi4 ,cube ,root 7,direct ,me?od74--,"o me?od ( extract+ ! cube root ( a numb] is by an inv]sion ( ! =ego+ op]ation4 ,us+ ! same scales1 ! slide is mov$ ei!r to ! "r or left until "u ! giv5 numb] on ,a is f\nd a numb] on ! ,l4,h4 ,b scale1 id5tical ) ! numb] simultane\sly f\nd on ,d "u ! "r or left index ( ,c4 ,?is numb] is ! requir$ cube root4 ,f :at has already be5 said reg>d+ ! comb9$ use ( ^! scales 9 cub+1 x w be evid5t t 9 extract+ ! cube root ( a numb]1 x is necess>y1 9 ord] to decide :i* scales are to be us$1 to "k ! numb] ( figures to be dealt )4 ,we !re=e 7z 9 ! ari?metical me?od ( extraction7 po9t (f ! giv5 numb] into sections ( ?ree figures ea*1 comm5c+ at ! decimal po9t1 & proce$+ to ! left = numb]s great] ?an unity1 & to ! "r = numb]s less ?an unity4 ,!n if ! fir/ section ( figures on ! left consi/s (-- #a figure1 ! numb] w evid5tly require to be tak5 on :at we h call$ ! 8units0 scale--i4e41 on ! ,l4,h4 scale ( ,a1 us+ ! ,l4,h4 index ( ,c4 ,if ( #b figures1 ! numb] w be tak5 on ! 8t5s0 scale--i4e41 on ! ,r4,h4 scale ( ,a1 us+ ! ,l4,h4 index ( ,c4 ,if ( #c figures1 ! numb] w be tak5 on ! 8hundr$s0 scale--i4e41 on ! ,l4,h4 scale ( ,a1 us+ ! ,r4,h4 index ( ,c4 ,to det]m9e ! numb] ( digits 9 cube roots x is only necess>y to note t :5 ! numb] is po9t$ (f into sections z direct$1 !re w be "o figure 9 ! root = e section into :i* ! numb] is s divid$1 :e!r ! fir/ section consi/s ( #a1 #b1 or #c digits4 ,( numb]s :olly decimal1 ! cube roots w be decimal1 & = e gr\p ( ?ree #js imm$iately foll[+ ! decimal po9t1 "o #j w foll[ ! decimal po9t 9 ! root4 ,if necess>y1 #js mu/ be add$ s z to make up complete multiples ( #c figures be=e proce$+ to extract ! root4 ,?us #jh is to be reg>d$ z #jhjj1 & #jjjjjh z #jjjjjhj 9 extract+ cube roots4 ,,ex4--,f9d 7#ad1jjj47 ,po9t+ ! numb] (f 9 ! mann] describ$1 x is se5 t !re are two figures 9 ! fir/ section--viz41 #ad4 ,sett+ ! cursor to #ad on ! ,r4,h4 scale ( ,a1 ! slide is mov$ to ! "r until x is se5 t #bda on ! ,l4,h4 scale ( ,b falls "u ! cursor1 :5 #bda on ,d is "u ! ,l4,h4 index ( ,c4 ,po9t+ #ad1jjj (f into sections we h #ad #jjj--t is1 two sections4 ,!re=e1 !re are two digits 9 ! root1 :i* 9 consequ5ce w be read #bda #bdajad4 ,,ex4--,f9d 7#jafb47 ,z ! divisional section consi/s ( ?ree figures1 we use ! 8hundr$s0 scale4 ,sett+ ! cursor to #jafb on ! ,l4,h4 ,a scale1 & us+ ! ,r4,h4 index ( ,c1 we move ! slide to ! left until "u ! cursor #jede is f\nd on ! ,l4,h4 ,b scale1 :ile ! ,r4,h4 index ( ,c po9ts to #jede on ,d1 :i* is !re=e ! cube root ( #jafb4 ,,ex4--,f9d 7#jjjjb47 ,to make ev5 multiples ( #c figures requires ! addition ( #jj2 we h !n #bjj1 ! cube root ( :i* is f\nd to be ab\t #ehe4 ,!n1 s9ce ! fir/ divisional gr\p consi/s ( #js1 "o #j w foll[ ! decimal po9t1 giv+ 7#jjjjb7 #jjehe #jjehdh4 ,cube ,root 7,inv]t$ ,slide ,me?od74--,ano!r me?od ( extract+ ! cube root involves ! use ( ! inv]t$ slide4 ,sev]al me?ods are us$1 b ! foll[+ is to be pref]r$3--,set ! ,l4,h4 or ,r4,h4 index ( ! slide to ! numb] on ,a1 & ! numb] on 7i4e41 ,b inv]t$71 :i* co9cides ) ! same numb] on ,d1 is ! requir$ root4 ,sett+ ! slide z direct$1 & us+ fir/ ! ,l4,h4 index ( ! slide & !n ! ,r4,h4 index1 x is always possible to f9d ?ree pairs ( co9cid5t values4 ,to det]m9e :i* ( ! ?ree is ! requir$ result is be/ %[n by an example4 ,,ex4--,f9d 7#e17 7#ej17 & 7#ejj47 ,sett+ ! ,r4,h4 index ( ! slide to #e on ,a1 x is se5 t #aga on ,d co9cides ) #aga on 4 ,!n sett+ ! ,l4,h4 index to #e on ,a1 fur!r co9cid5ces are f\nd at #cfh & at #gic1 ! ?ree values ?us f\nd be+ ! requir$ roots4 ,note t ! fir/ root 0 f\nd on t portion ( ! ,d scale ly+ "u #a to #e on ,a2 ! second root on t portion ly+ "u #e to #ej on ,a2 & ! ?ird root on t portion ( ,d ly+ "u #ej to #ajj on ,a4 ,9 ?is connection1 !re=e1 scale ,a may always be consid]$ to be divid$ into ?ree sections--viz41 #a to n1 n to #ajn1 & #ajn to #ajj4 ,= all numb]s consi/+ ( #a1 #a #c1 #a #f1 #a #i--;i4e41 ( #a1 #d1 #g1 #aj1 or #b1 #e1 etc41 figures--! co9cid5ce "u ! fir/ section is ! "o requir$4 ,if ! numb] has #b1 #e1 #h1 or #a1 #d1 #g1 etc41 figures1 ! co9cid5ce "u ! second section is correct1 :ile if ! numb] has #c1 #f1 #i1 or #j1 #c1 etc41 figures1 ! co9cid5ce "u ! la/ section is t requir$4 ,! numb] ( digits 9 ! root is det]m9$ by m>k+ (f ! numb] into sections1 z already expla9$4 ,cube ,root 7,pickwor?'s ,me?od74--,"o ( ! pr9cipal objections to ! two me?ods describ$ is ! difficulty ( recollect+ :i* scales are to be employ$ & ) :i* index ( ! slide !y are to be us$4 ,) ! direct me?od ano!r objection is t ! read+s to be comp>$ are (t5 "s di/ance ap>t1 ! maximum di/ance int]v5+ be+ two-?irds ( ! l5g? ( ! rule4 ,to c>ry ! eye f "o to ano!r is tr\blesome & "t-tak+4 ,) ! inv]t$ scale me?od ! read+ ( a scale rev]s$ 9 direction & ) ! figures inv]t$ is also objectionable4 ,) ! au?or's me?od ^! objections are entirely obviat$4 ,! same scales & index are always us$1 & are read 9 _! normal position4 ,! ?ree roots ( n1 #ajn & #ajjn 7n be+ less ?an #aj & n less ?an #a7 are giv5 ) "o sett+ & appe> 9 _! natural sequ5ce1 no trav]s+ ( ! slide be+ ne$$4 ,! read+s to be comp>$ are always close toge!r1 ! maximum di/ance betwe5 !m be+ "o-six? ( ! l5g? ( ! rule4 ,! sett+ is always made 9 ! e>li] "p ( ! scales :]e clos] read+s c be obta9$1 & f9ally1 if desir$1 ! result may be readily v]ifi$ on ! l[] scales by successive multiplication4 ,= ?is me?od two gauge po9ts are requir$ on ,c4 ,to conv5i5tly locate ^!1 set #ec on ,c to #bdf on ,d2 jo9 #a on ,d to #a on ,a ) a /raight-edge & ) a ne$le po9t draw a %ort f9e l9e on ,c4 ,set #bdf on ,c to #ec on ,d1 & repeat ! process at ! o!r end ( ! rule4 ,! gauge po9ts ?us obta9$ 7divid+ ,c into ?ree equal p>ts7 w be at #baed & #dfda1 & %\ld be m>k$ 7#aj7 & 7#ajj7 respectively4#f ,,ex4--,f9d 7#bhf17 7#bhf7 & 7#bhf74 ,set cursor to #bhf on ,a & draw+ ! slide to ! "r f9d #adb "u #a on ,c1 :5 #adb on ,b is "u ! cursor4 ,!n read+ "u #a1 7#aj7 & 7#ajj17 we h 7#bhf7 #adb2 7#bhf7 #cjf & 7#bhf7 #fei4 ,x w be se5 t factoris+ ) p[]s ( #aj1 we multiply ! initial root by 7#aj7 & 7#ajj74 ,obvi\sly ! ?ree roots w always be f\nd on ,d1 9 _! natural ord] & at int]vals ( "o-?ird ! l5g? ( ! rule4 ,! numb] ( digits 9 ! roots ( numb]s :i* d n lie betwe5 #a & #ajjj1 is f\nd z be=e expla9$4 ,9 any me?od ( extract+ cube roots 9 :i* ! slide has to be adju/$ to give equal read+s on ,b & ,d1 ! au?or has f\nd x ( advantage to adopt ! foll[+ plan3--,! cursor be+ set to1 say1 #dh on ,a1 br+ a ne> ma9 division l9e on ,b1 z #ag1 to ! cursor2 !n #a on ,c is at #afh on ,d4 ,! diff]5ce 9 ! read+s is two small divisions on ,d1 & mov+ ! slide =w>d by "o-?ird ! space repres5t+ ?is diff]5ce1 we obta9 #afhg z ! root requir$4 ,) a little practice x is possible to obta9 m accurate results by ?is me?od ?an by comp>+ ! read+ on ,d ) t on ! less f9ely-grad$ ,b scale4 ,,miscellane\s ,,p[]s ,,& ,,roots4 ,9 addition to squ>es & cubes1 c]ta9 o!r p[]s & roots may be readily obta9$ ) ! slide rule4 ,two-?irds ,p[]4--,! value ( ,n is f\nd on ,a ov] ,n on ,d4 ,! numb] ( digits is decid$ by ! rule = squ>es1 work+ f ! numb] ( digits 9 ! cube root4 ,x w (t5 be f\nd pref]able to treat ,n z ,n ,n1 z 9 ?is way ! magnitude ( ! result is mu* m readily appreciat$4 ,?ree-two ,p[]4--,n c be obta9$ by cub+ ! squ>e root1 decid+ ! numb] ( digits 9 ea* process4 ,= ! reason j giv51 x is pref]able to reg>d ,n z ,n ,n4 ,f\r? ,p[]4--,= ,n#d set ! index ( ,c to ,n on ,d & ov] ,n on ,c read ,n#d on ,a2 or f9d ! squ>e ( ! squ>e ( ,n1 decid+ ! numb] ( digits at ea* /ep4 ,f\r? ,root4--,simil>ly = ,n1 take ! squ>e root ( ! squ>e root4 ,f\r-?ird ,p[]4--,n ,n#acc 7useful 9 gas-eng9e diagram calculations7 is be/ treat$ z ,n ,n4 ,o!r p[]s c be f\nd by repeat$ multiplication4 ,?us sett+ #a on ,b to ,n on ,a1 we h on ,a1 ,n#b ov] ,n2 ,n#c ov] ,n#b2 ,n#d ov] ,n#c2 ,n#e ov] ,n#d1 etc4 ,9 ! same way1 sett+ ,n on ,b to ,n on ,d1 we c read su* values z ,n1 ,n1 etc4 ,,p[]s ,,& ,,roots ,,by ,,log>i?ms4 ,= p[]s or roots o!r ?an ^? ( ! simple =ms already discuss$1 x is necess>y to employ ! usual log>i?mic process4 ,?us to f9d an x1 we multiply ! log>i?m ( a by n1 & f9d ! numb] x correspond+ to ! log>i?m s obta9$4 ,simil>ly1 to f9d a x we divide ! log>i?m ( a by n1 & f9d ! numb] x correspond+ to ! result+ log>i?m4 ,! ,scale ( ,log>i?ms4--,^u ! back ( ! slide ( ! ,gravt & simil> slide rules !re w be f\nd ?ree scales4 ,"o ( ^!--usually ! c5tre "o--is divid$ equally ?r\gh\t its entire l5g?1 & figur$ f "r to left4 ,x is sometimes m>k$ ,l1 indicat+ t x is a scale giv+ log>i?ms4 ,! :ole scale is divid$ prim>ily into t5 equal p>ts1 & ea* ( ^! subdivid$ into #ej equal p>ts4 ,9 ! recess or not* 9 ! "r-h& end ( ! rule is a ref]5ce m>k1 to :i* any ( ! divisions ( ?is ev5ly-divid$ scale c be set4 ,z ?is decimally-divid$ scale is equal 9 l5g? to ! log>i?mic scale ,d1 & is figur$ 9 ! rev]se direction1 x results t :5 ! slide is drawn to ! "r s t ! ,l4,h4 index ( ,c co9cides ) any numb] on ,d1 ! read+ on ! equally-divid$ scale w give ! decimal "p ( ! log>i?m ( ! numb] tak5 on ,d4 ,?us if ! ,l4,h4 index ( ,c is plac$ to agree ) #b on ,d1 ! read+ ( ! back scale1 tak5 at ! ref]5ce m>k1 w be f\nd to be #jcja1 ! log>i?m ( #b4 ,x mu/ be di/9ctly borne 9 m9d t ! numb] s obta9$ is ! decimal "p or mantissa ( ! log>i?m ( ! numb]1 & t to ?is ! *>act]i/ic mu/ be prefix$ 9 accordance ) ! usual rule--viz41 ,! integral "p1 or *>act]i/ic ( a log>i?m is equal to ! numb] ( digits 9 ! numb]1 m9us #a4 ,if ! numb] is :olly decimal1 ! *>act]i/ic is equal to ! numb] ( cyph]s foll[+ ! decimal po9t1 plus #a4 ,9 ! latt] case ! *>act]i/ic is negative1 & is s indicat$ by hav+ ! m9us sign writt5 ov] x4 ,to obta9 any giv5 p[] or root ( a numb]1 ! op]ation is z foll[s3--,set ! ,l4,h4 index ( ,c to ! giv5 numb] on ,d1 & turn+ ! rule ov]1 read opposite ! m>k 9 ! not* at ! "r-h& end ( ! rule1 ! decimal "p ( ! log>i?m ( ! numb]4 ,add ! *>act]i/ic accord+ to ! above rule1 & multiply by ! expon5t ( ! p[]1 or divide by ! expon5t ( ! root4 ,place ! decimal "p ( ! resultant read+1 tak5 on ! scale ( equal p>ts1 opposite ! m>k 9 ! ap]ture ( ! rule1 & read ! answ] on ,d "u ! ,l4,h4 index ( ,c1 po9t+ (f ! numb] ( digits 9 ! answ] 9 accordance ) ! numb] ( ! *>act]i/ic ( ! resultant4 ,,ex4--,evaluate #cfadad4 ,set #a on ,c to #cf on ,d & read ! decimal "p ( log4 #cf on ! scale ( log>i?ms on ! back ( ! slide4 ,?is value is f\nd to be #jeef4 ,z !re are two digits 9 ! numb]1 ! *>act]i/ic w be #a2 h5ce log4 #cf #aeef4 ,multiply by #adad1 us+ ! ,c & ,d scales1 & obta9 #bb z ! log4 ( ! result4 ,set ! decimal "p1 #jb1 on ! log4 scale to ! m>k 9 ! not* at ! end ( ! rule & read #aehe on ,d "u #a on ,c4 ,s9ce ! log4 ( ! result has a *>act]i/ic #b1 !re w be #c digits 9 ! result1 :i* is !re=e read z #aehe4 ,?is example w suffice to %[ ! me?od ( obta9+ ! n? p[] or ! n? root ( any numb]4 ,,o!r ,,me?ods ,,( ,,obta9+ ,,p[]s ,,& ,,roots4 ,a simple me?od ( obta9+ p[]s & roots1 :i* may s]ve on occasion1 is by scal+ (f proportional l5g?s on ! ,d scale 7or ! ,a scale7 ( ! ord9>y rule4 ,?us1 to det]m9e ! value ( #abeafg we take ! actual l5g? #a-abe on ,d scale1 & increase x by any conv5i5t means 9 ! proportion ( #a #afg4 ,!n ) a pair ( divid]s we set (f ?is new l5g? f #a1 & obta9 #add z ! result4 ,"o conv5i5t me?od ( obta9+ ! desir$ ratio is by a pair ( proportional compasses4 ,?us to obta9 #aeb1 ! compasses w\ld be set 9 ! ratio ( #af to #ag1 & ! small] end op5$ out to include #a-aeb on ! ,d scale2 ! op5+ 9 ! l>ge end ( ! compasses w !n be su* t sett+ x (f f #a we obta9 #aef on ,d z ! result s\ght4 ,illu/ration3 ,,fig4 #aa4 ,! conv]se proc$ure = obta9+ ! n? root ( a numb] ,n w obvi\sly resolve itself into obta9+ 7#a7_/7n7? ( ! scale l5g? #a-,n1 & ne$ n be fur!r consid]$4 ,simple geometrical con/ructions are also us$ = obta9+ scale l5g?s 9 ! requir$ ratio4 ,a s]ies ( p>allel l9es rul$ on transp>5t celluloid or /\t trac+ pap] may be plac$ 9 an incl9$ position on ! face ( ! rule & adju/$ s z to divide ! scale z desir$4 ,:5 mu* "w is to be done :i* requires values to be rais$ to "s con/ant b comp>atively l[ p[]1 n1 ! au?or has f\nd ! foll[+ device ( assi/ance3--,on a piece ( ?9 transp>5t celluloid a l9e ,,oc is drawn 7,fig4 #aa7 & 9 ?is a po9t ,b is tak5 su* t 7,,oc7_/7,,ob7 is ! desir$ ratio4 ,x is conv5i5t to make ,,ob #a-aj on ! ,a scale1 s t assum+ we require a s]ies ( values ( v#ace1 ,,ob w\ld be #abe cm4 & ,,oc1 #afhge cm4 ,on ^! l9es semi-circles are drawn z %[n1 bo? pass+ ?r\gh ! po9t ,o4 ,apply+ ?is cursor to ! upp] scales s t ! po9t ,o is on #a & ! semi-circle ,o ,m ,b passes ?r\gh v on ,a1 ! l>g] semi-circle w give on ,a ! value ( vn4 ,?us = p vn #cie #diace1 set #a on ,b to #cie on ,a 7,fig4 #ab7 & apply ! cursor to ! work+ edge ( ,b1 s t ,o agrees ) #a & ,o ,m ,b passes ?r\gh #di on ,b4 ,! l>g] semi-circle !n cuts ! edge ( ! slide on a po9t1 giv+ #ccg on ,a z ! result requir$4 ,( c\rse any numb] ( semi-circles may be drawn1 giv+ diff]5t ratios4 ,if a numb] ( ev5ly-spac$ divisions are us$ z bases1 ! device af=ds a simple means ( obta9+ a succession ( small p[]s or roots1 :ile x also f9ds a use 9 det]m9+ a numb] ( geometric means betwe5 two values z is requir$ 9 arrang+ ! spe$ ge>s ( ma*9e tools1 etc4 ,! conv]se op]ation ( f9d+ roots w be evid5t z w also _m o!r uses = :i* ! device is ( s]vice4 ,illu/ration3 ,,fig4 #ab4 ,! l9es %\ld be drawn 9 ,indian ink ) a v %>p p5 & on ! "u side ( ! celluloid s t ! l9es lie 9 close contact ) ! face ( ! rule4 ,! ,radial ,cursor1 ano!r device = ! same purpose1 is always us$ 9 conjunction ) ! upp] scales4 ,z w be se5 f ,fig4 #ac1 ! body ( ! cursor ,p c>ries a graduat$ b> ,s :i* c be remov$ 9 a direction transv]se to ! rule1 & adju/$ to any desir$ position4 ,pivot$ to ! l[] end ( ,s is a radial arm ,r ( transp>5t celluloid on :i* a c5tre l9e is engrav$4 ,a ref]5ce to ! illu/ration w %[ t ! pr9ciple involv$ is t ( simil> triangles1 ! wid? ( ! slide be+ us$ z "o ( ! elem5ts4 ,?us1 to take a simple case1 if #b on ,s is set to ! index on ,p1 & #a on ,b is br\ght to ,n on ,a1 !n by sw++ ! radial arm until its c5tre l9e agrees ) #a on ,c1 we c read ,n#b on ,a4 ,evid5tly1 s9ce 9 ! two simil> triangles ,a ,o ,n#b & ,n t ,n#b ! l5g? ( ,a ,o is twice t ( ,n t1 x results t ,a ,n#b #b ,a ,n4 ,9 g5]al1 !n1 to f9d ! n? p[] ( a numb]1 we set ! cursor to #a or #aj on ,a1 br+ n on ! cross b> ,s to ! index on ! cursor1 & #a on ,b to ,n on ,a4 ,!n to #a on ,c we set ! l9e on ! radial arm1 & "u ! latt] read ,nn on ,a4 ,! inv]se proce$+ = f9d+ ! n? root w be obvi\s4 ,illu/ration3 ,,fig4 #ac4 ,an advantage (f]$ by ?is & analog\s me?ods ( obta9+ p[]s & roots is t ! result is obta9$ on ! ord9>y scale ( ! rule1 & h5ce x c be tak5 directly into any fur!r calculation :i* may be necess>y4 ,,comb9$ ,,op]ations4 ,?us f> ! v>i\s op]ations h be5 sep>ately consid]$1 & we n[ pass on to a consid]ation ( ! me?ods ( work+ = solv+ ! v>i\s =mul met ) 9 te*nical calculations4 ,we propose to expla9 ! me?ods ( deal+ ) a few ( ! m g5]ally us$ expressions1 z ?is w suffice to sugge/ ! proc$ure 9 deal+ ) o!r & m intricate calculations4 ,9 solv+ ! foll[+ problems1 bo? ! upp] & l[] scales are us$1 & ! relative value ( ! sev]al scales mu/ be obs]v$ ?r\gh\t4 ,?us1 9 solv+ su* an expression z 77#gde7_/7aeh77 #fhf1 ! division is fir/ effect$ by sett+ #aeh on ,b to #gde on ,a4 ,f ! relation ( ! two p>ts ( ! upp] scales 7page #cg7 we "k t su* values z #gde1 #gde1 etc41 w be tak5 on ! left-h& ,a & ,b scales1 :ile values z #aeh1 #aehj1 etc41 w be tak5 on ! "r-h& ,a & ,b scales4 ,h5ce1 #aeh on ! ,r4,h4 ,b scale is set to #gde on ! ,l4,h4 ,a scale1 & ! result read on ,d "u ! index ( ,c4 ,_h bo? values be5 tak5 on ! ,l4,h4 ,a & ,b scales1 or bo? on ! ,r4,h4 ,a & ,b scales1 ! results w\ld h correspond$ to x 77#gde7_/7aeh77 #bag1 or to x 77#gde7_/7aeh77 #bag1 i4e41 to 7#fhf7_/77aj774 ,h5ce if a wrong *oice ( scales has be5 made1 we c correct ! result by multiply+ or divid+ by 7#aj7 z ! case may require4 ,if ! result is read on ,d1 set to x ! c5tre index 7#aj7 ( ,b & read ! correct$ result "u ! index ( ,c4 ,to solve a b#b x4 ,set ! index ( ,c to b on ,d1 & ov] a on ,b read x on ,a4 ,to solve 7a#b7_/7;b7 x4 ,set b on ,b to a on ,d by us+ ! cursor1 & ov] index ( ,b read x on ,a4 ,to solve 7b7_/7a#b7 x4 ,set a on ,c to b on ,a1 & ov] #a on ,b read x on ,a4 ,to solve 7a b#b7_/7;c7 x4 ,set c on ,b to b on ,d1 & ov] a on ,b read x on ,a4 ,to solve 7a b7#b x4 ,set #a on ,c to a on ,d1 & ov] b on ,c read x on ,a4 ,to solve 77a7_/7b77#b x4 ,set b on ,c to a on ,d1 & ov] #a on ,c read x on ,a4 ,to solve 7a b7 x4 ,set #a on ,b to a on ,a1 & "u b on ,b read x on ,d4 ,to solve 77a7_/7b77 x4 ,set b on ,b to a on ,a1 & "u #a on ,c read x on ,d4 ,to solve a 7b7_/7c#b7 x4 ,set b on ,c to c on ,d & ov] a on ,b read x on ,a4 ,to solve c77a7_/7b77 x4 ,set b on ,b to a on ,a1 & "u c on ,c read x on ,d4 ,to solve 7a7_/7b7 x4 ,set b on ,c to a on ,a1 & "u #a on ,c read x on ,d4 ,to solve 7a7_/7b7 x4 ,set b on ,b to a on ,d1 & "u #a on ,c read x on ,d4 ,to solve ba x4 ,set #a on ,c to b on ,d1 & "u a on ,b read x on ,d4 ,to solve 7a#c7 x4 ,treat z aa4 ,to solve a7b#c7 x4 ,treat z ab b4 ,to solve 7a#c7_/7;b7 x4 ,treat z 7a a7_/7b74 ,to solve 77a#c7_/7;b77 x4 ,treat z 7a a7_/7b7 77a7_/7b77 a4 ,to solve 77a b7_/7c77 x4 ,set c on ,b to a on ,a1 & "u b on ,b read x on ,d4 ,to solve 7a b7_/7c7 x4 ,set c on ,b to b an ,d1 & "u a on ,c read x on ,d4 ,to solve 77a#b b7_/7c77 x4 ,set c on ,b to a on ,d1 & "u b on ,b read x on ,d4 ,to solve 7a#b b#b7_/7;c7 x4 ,set c on ,b to a on ,d1 & ov] b on ,c read x on ,a4 ,to solve 7ab7_/7c7 x4 ,set c on ,c to b on ,a1 & "u a on ,c read x on ,d4 ,to solve 77a b7_/7c77#b x4 ,set c on ,c to a on ,d1 & ov] b on ,b read x on ,a4 ,,h9ts ,,on ,,evaluat+ ,,expressions4 ,z a g5]al rule1 ! use ( cubes & high] p[]s %\ld be avoid$ :5ev] possible4 ,?us1 9 ! =ego+ section1 we recomm5d treat+ an expression ( ! =m a7b#c7 z a b b2 ! magnitudes ( ! values ?us met ) are m easily appreciat$ by ! beg9n]1 & mi/akes 9 e/imat+ ! l>ge numb]s involv$ 9 cub+ are avoid$4 ,,ex4--#gc 7#egc7 #cadj4 ,set #a on ,c to #eg on ,d2 br+ cursor to #eg on ,b 7,r4,h41 s9ce #eg has an ev5 numb] ( digits72 br+ #a on ,c to cursor1 & "u #gc on ,c read #cadj on ,d4 ,z a r\gh e/imate we h 7#eg71 ab\t #h2 #h #eg1 ab\t #djj2 #djj #g1 gives #bhjj1 %[+ ! result consi/s ( #d figures4 ,an expression ( ! =m a7b#b71 or a b1 is bett] dealt ) by re>rang+ z a 7b7_/7b74 ,,ex4--#cfd7dcbb7 #ife4 ,set cursor to #dcb on ,a1 & move ! slide until #afc is f\nd simultane\sly "u ! cursor on ,b & on ,d "u #a on ,c2 br+ cursor to #a on ,c2 #dcb on ,c to cursor1 & ov] #cfd on ,d read #ife on ,c4 7,note t 9 ?is case x is conv5i5t to read ! answ] on ! slide2 see page #bb74 ,f ! slide rule we "k 7#dcb7 ab\t #af2 ?is into #dcb is r\ghly #c2 #cfd #c is ab\t #aj1 %[+ ! answ] to be #ife4 ,simil>ly products ( ! =m a b are be/ dealt ) z a b b4 ,factoris+ expressions sometimes simplifies matt]s1 z1 = in/ance1 9 x#d y#d 7x#b y#b77x#b y#b74 ,"h1 work+ ) ! f\r? p[]s involves l>ge numb]s & ! tr\blesome det]m9ation ( ! numb] ( digits 9 ea* factor2 b squ>es are read on ! rule at once1 ! numb] ( digits is obvi\s1 &1 9 g5]al1 ! me?od %\ld give a m accurate result4 ,take ! expression1 ,d#a 77,d#d d#d7_/7;,d77 giv+ ! diamet] ,d#a ( a solid %aft equal 9 torsional /r5g? to a holl[ %aft ^: ext]nal & int]nal diamet]s are ,d & d respectively4 ,re>rang+ z ,d#a 777,d#b d#b77;,d#b d#b77_/7;,d77 & tak+1 z an example1 ,d #ae 94 & d #g 941 we h ,d#b d#b #bgd & ,d#b d#b #agf2 h5ce ,d#a 77#bgd #agf7_/7ae77 7#cbaj7 #adge 94 ,rev]s$ ,scale ,notation4--,) expressions ( ! =m #a x1 or #ajj x1 x is (t5 conv5i5t to reg>d ! scales z hav+ _! notation rev]s$1 i4e41 to read ! scale backw>ds4 ,:5 ?is is done ! ,d scale is read z %[n on ! l[] l9e-- ,direct ,notation #a #b #c #d #e #f #g #h #i #aj ,d ,scale ,rev]s$ ,notation #i #h #g #f #e #d #c #b #a #j ,! new read+ c be f\nd by subtract+ ! ord9>y read+ f #a1 #aj1 #ajj1 etc41 accord+ to ! value assign$ to ! ,r4,h4 index1 b actually x is unnecess>y to make ?is calculation1 z ) a little practice x is q an easy matt] to read bo? ! ma9 & subdivisions 9 ! rev]s$ ord]4 ,applications are f\nd 9 plott+ curves1 trigonometrical =mul1 etc4 ,,ex4--,f9d ! p] c5t4 ( slip ( a screw propell] f #ajj ,s 7#ajacc,v7_/7,,pr7 tak+ ! spe$1 ,v1 z #ae knots1 ! pit* ( ! propell]1 ,p1 z #bg ft4 #f 941 & ! revolutions p] m9ute1 ,r1 z #fj4 ,set #bge on ,b to #ajacc on ,a 7,n4,b4--,take ! sett+ ne> ! c5tre index ( ,a72 br+ ! cursor to #ae on ,b & #fj on ,b to cursor4 ,read+ ! ,l4,h4 ,a scale backw>ds1 ! slip1 ,s1 #h p] c5t4 is f\nd on ,a ov] #aj on ,b4 ,p]c5tage ,calculations4--,to increase a quantity by x p] c5t4 we multiply by #ajj x2 to dim9i% a quantity by x p] c5t4 we multiply by #ajj x4 ,h5ce1 to add x p] c5t41 set #ajj x on ,c to #a on ,d & read new values on ,d "u orig9al values on ,c4 ,to d$uct x p] c5t4 read ! ,d scale backw>ds f #aj & set ,r4,h4 index ( ,c to x p] c5t4 s read4 ,!n read z be=e4 ,,gauge ,,po9ts4 ,special graduations1 m>k+ ! position ( con/ant factors :i* frequ5tly ent] into eng9e]+ calculations1 are f\nd on mo/ slide rules4 ,usually ! values ( #cadaf & 77_/7#d7 #jghed--! 8gauge po9ts0 = calculat+ ! circumf]5ce & area ( a circle--are m>k$ on ! upp] scales4 ,! fir/ %\ld be giv5 on ! l[] scales also4 ,m>ks c & c#a are sometimes f\nd on ! l[] scales at #aabh 77#d7_/777 & at #cefh 77#dj7_/7774 ,^! are useful 9 calculat+ ! cont5ts ( cyl9d]s & are ?us d]iv$3--,cubic cont5ts ( cyl9d] ( diamet] d & l5g? l 77_/7#d7;d#bl2 sub/itut+ = 77_/7#d7 its reciprocal 7#d7_/771 ! =mula becomes 7d#b7_/7abgc l71 & by tak+ ! squ>e root ( ! fractional "p we h 7d7_/7#aabh7b l4 ,?is is n[ 9 a v conv5i5t =m1 s9ce by sett+ ! gauge po9t c on ,c to d on ,d1 we c read ov] l on ,b ! cubic cont5ts on ,a4 ,?is example indicates ! pr9ciple to be foll[$ 9 arrang+ gauge po9ts4 ,successive multiplication is avoid$ by sub/itut+ ! reciprocal ( ! con/ant1 ?us br++ ! expression into ! =m 7a b7_/7c71 :i*1 z we "k1 c be resolv$ by "o sett+ ( ! slide4 ,! advantage ( divid+ d be=e squ>+ is also evid5t4 ,! m>k c#a c 7#aj7 is us$ if x is necess>y to draw ! slide m ?an "o-half its l5g? to ! "r4 ,a gauge po9t1 ,m1 at #cahc 7#ajj7_/77 is f\nd on ! upp] scales ( "s rules4 ,sett+ ?is po9t on ,b to ! diamet] ( a cyl9d] on ,a1 ! circumf]5ce is read ov] #a or #ajj on ,b or ! area ( ! curv$ surface ov] ! l5g? on ,b4 ,z ano!r example ( e/abli%+ a gauge po9t1 we w take ! =mula = ! !oretical deliv]y ( pumps4 ,if d is ! diamet] ( ! plung] 9 in*es1 l ! l5g? ( /roke 9 feet1 & ,q ! deliv]y 9 gallons1 we h ,q d#b 77_/7#d7 l 7#ab7_/7bgg74 7,n4,b4--#bgg cubic in*es #a gallon47 ,multiply+ out ! con/ant quantities & tak+ its reciprocal1 we readily trans=m ! /atem5t into ,q 7d#bl7_/7#bid7 or 77d7_/7#edb77b l4 ,h5ce set gauge po9t #edb on ,c to d on ,d & ov] l5g? ( /roke 9 feet on ,b1 read deliv]y 9 gallons p] /roke on ,a2 or ov] pi/on spe$ 9 feet p] m9ute on ,b1 read !oretical deliv]y 9 gallons p] m9ute on ,a4 ,sev]al examples ( gauge po9ts w be f\nd 9 ! section on calculat+ ! weights ( metal 7see pages #ei & #fj74 ,9 mo/ cases _! d]ivation w be evid5t f :at has be5 said above4 ,9 ! case ( ! weight ( sph]es1 we h ,vol4 #jebcf;d#c1 & ?is multipli$ by ! weight ( #a cubic in* ( ! mat]ial w give ! weight ,w 9 lb4 ,h5ce = ca/-iron1 ,w #jebcf d#c #jbf1 :i* is conv5i5tly trans=m$ into ,w 7d d#b7_/7gce7 z 9 ! example on page #fj4 ,) ^! examples no difficulty %\ld be exp]i5c$ 9 e/abli%+ gauge po9ts = any calculation 9 :i* con/ant factors recur4 ,m>k+ ,gauge ,po9ts4--,! practice ( m>k+ gauge po9ts by l9es ext5d+ to ! work+ edge ( ! scale is n to be recomm5d$1 z x confuses ! ord9>y read+ ( ! scales4 ,g5]ally speak+1 gauge po9ts are only requir$ occasionally1 & if !y are plac$ cle> ( ! scale to :i* !y p]ta91 b ne> en\gh to %[ ! connection1 !y c be br\ght readily into a calculation by means ( ! cursor4 ,usually !re is suffici5t m>g9 above ! ,a scale & bel[ ! ,d scale = v>i\s gauge po9ts to be m>k$4 ,ano!r plan consi/s 9 cutt+ two nicks 9 ! upp] & l[] edges ( ! cursor ne> ! c5tre & ab\t 94 ap>t4 ,^! c5tre pieces1 :5 b5t out1 =m a tongue1 :i* are 9 l9e ) ! cursor l9e & run ne>ly 9 contact ) ! squ>e & bevell$ edges ( ! rule respectively4 ,a f9e l9e 9 ! tongue c !n be set to gauge po9ts m>k$ on ^! two edge /rips1 ! ord9>y measur+ graduations be+ remov$1 if desir$1 by a piece ( f9e s&-pap]4 ,= gauge po9ts m>k$ on ! face ( ! rule1 ! au?or pref]s two f9e l9es drawn at #de--?us1 --& cross+ 9 ! exact po9t :i* x is requir$ to indicate4 ,) ! 8cross0 gauge po9t ! meet+ l9es facilitate ! plac+ ( ! cursor1 & an exact sett+ is readily made4#g ,all l9es %\ld be drawn 9 ,indian ink ) a v %>p draw+ p54 ,= a m p]man5t m>k+ ! ,indian ink may be rubb$ up 9 glacial acetic acid or ! special ink = celluloid may be us$4 ,if any difficulty is f\nd 9 writ+ ! di/+ui%+ signs aga9/ ! gauge po9t1 ! inscription may be =m$ by a succession ( small dots made ) a %>p prick]4 ,,examples ,,9 ,,te*nical ,,calculations4 ,9 ord] to illu/rate ! practical value ( ! slide rule1 we n[ give a numb] ( examples :i* w d\btless be suffici5t to sugge/ ! me?ods ( work+ ) o!r =mul4 ,a few ( ! rules give results :i* are approximate only1 b 9 all cases ! degree ( accuracy obta9$ is well )9 ! possible read+ ( ! scales4 ,9 _m cases ! rules giv5 may be modifi$1 if desir$1 by v>y+ ! con/ants4 ,9 mo/ ( ! examples ! p>ticul> =mula employ$ w be evid5t f ! solution1 b 9 a few ( ! m complicat$ cases a sep>ate /atem5t has be5 giv54 ,,m5suration1 ,,etc4 ,giv5 ! *ord c ( a circul> arc1 & ! v]tical height h1 to f9d ! diamet] d ( ! circle4 ,set ! height h on ,b to half ! *ord on ,d1 & ov] #a on ,b read x on ,a4 ,!n x h d4 ,,ex4--c #f2 h #b2 f9d d4 ,set #b on ,b to #c on ,d1 & ov] #a on ,b read #de on ,a4 ,!n #de #b #fe d4 ,giv5 ! radius ( a circle r1 & ! numb] ( degrees n 9 an arc1 to f9d ! l5g? l ( ! arc4 ,set r on ,c to #egc on ,d1 & ov] any numb] ( degrees n on ,d read ! 7approximate7 l5g? ( ! arc on ,c4 ,,ex4--r #bd2 n #cj2 f9d l4 ,set #bd on ,c to #egc on ,d1 & ov] #cj on ,d read #abef l on ,c4 ,giv5 ! diamet] d ( a circle 9 in*es1 to f9d ! circumf]5ce c 9 feet4 ,set #aia on ,c to #ej on ,d1 & "u any diamet] 9 in*es on ,c read circumf]5ce c1 9 feet on ,d4 ,,ex4--,f9d ! circumf]5ce 9 feet ( a pulley #ag 94 9 diamet]4 ,set #aia on ,c to #ej on ,d1 & "u #ag on ,c read #dde ft4 on ,d4 ,giv5 ! diamet] ( a circle1 to f9d its area4 ,set #jghed on ,b to #aj 7c5tre index7 on ,a & ov] any diamet] on ,d read area on ,b4 ,:5 ! rule has a special graduation l9e #jghed1 on ! "r-h& scale ( ,b1 set ?is l9e to ! ,r4,h4 index ( ,a & read (f z above4 ,if only is m>k$1 set ?is special graduation on ,b to #d on ,a4 ,on ! ,c & ,d scales ( "s rules a gauge po9t m>k$ c w be f\nd indicat+ 77#d7_/777 #aabhf4 ,9 ?is case1 !re=e1 set #a on ,c to gauge po9t c on ,d1 & read area on ,a z above4 ,if ! gauge po9t c is us$1 divide ! result by #aj4 ,or set c on ,c1 to diamet] on ,d1 & ov] index ( ,b read area on ,a4 ,cursors are suppli$1 hav+ two l9es rul$ on ! glass1 ! int]val betwe5 !m be+ equal to 7#d7_/77 #abgc on ! ,a scale4 ,9 ?is case1 if ! "r h& ( ! two cursor l9es be set to ! diamet] on ,d1 ! area w be read on ,a "u ! left-h& cursor l9e4 ,= diamet]s less ?an #aaa x is necess>y to set ! middle index ( ,b to ! ,l4,h4 index ( ,a1 read+ ! areas on ! ,l4,h4 ,b scale4 ,! confusion :i* 9 g5]al "w is sometimes caus$ by ! use ( two cursor l9es might be obviat$ by mak+ ! left-h& l9e 9 two %ort l5g?s1 ea* only j cov]+ ! scales4 ,giv5 diamet] ( circle d 9 in*es1 to f9d area a 9 squ>e feet4 ,set #f on ,b to #aa on ,a1 & ov] diamet] 9 in*es on ,d read area 9 squ>e feet on ,b4 ,to f9d ! surface 9 squ>e feet ( boil] flues1 cond5s] tubes1 heat+ pipes1 etc41 hav+ giv5 ! diamet] 9 in*es & l5g? 9 feet4 ,f9d ! circumf]5ce 9 feet z above & multiply by ! l5g? 9 feet4 ,,ex4--,f9d ! heat+ surface af=d$ by #afj locomotive boil] tubes #a;94 9 diamet] & #ab ft4 long4 ,set #aia on ,c to #ej on ,d2 br+ cursor #age on ,c1 ,l4,h4 index ( ,c to cursor2 cursor to #ab on ,c2 #a on ,c to cursor2 & "u #afj on ,c read #hhj sq4 ft4 ( heat+ surface on ,d4 ,if ! dim5sions are 9 ! same d5om9ation & ! rule has a gauge po9t ,m at #cahc 7 7#ajj7_/7771 set ?is m>k on ,b to diamet] ( cyl9d] on ,a1 & read cyl9drical surface on ,a ov] l5g? on ,b4 ,to f9d ! side s ( a squ>e1 equal 9 area to a giv5 rectangle ( l5g? l & bread? b4 ,set ,r4,h4 or ,l4,h4 index ( ,b to l on ,a1 & "u b on ,b read s on ,d4 ,,ex4--,f9d ! side ( a squ>e equal 9 area to a rectangle 9 :i* l #ca ft4 & b #e ft4 ,set ! 7,r4,h47 index ( ,b to #ca on ,a1 & "u #e on ,b read #abde ft4 on ,d4 ,to f9d v>i\s l5g?s l & bread?s b ( a rectangle1 to give a con/ant area a4 ,inv]t ! slide & set ! index ( to ! giv5 area on ,d4 ,!n opposite any l5g? l on f9d ! correspond+ bread? b on ,d4 ,,ex4--,f9d ! correspond+ bread?s ( rectangul> %eets1 #af1 #ah1 #bd1 #cf1 & #fj ft4 long1 to give a con/ant area ( #gb sq4 ft4 ,set ! ,r4,h4 index ( to #gb on ,d1 & opposite #af1 #ah1 #bd1 #cf1 & #fj on read #de1 #d1 #c1 #b1 & #ab ft41 ! correspond+ bread?s on ,d4 ,to f9d ! cont5ts 9 cubic feet ( a cyl9d] ( diamet] d 9 in*es & l5g? l 9 feet4 ,f9d area 9 feet z be=e1 & multiply by ! l5g?4 ,if dim5sions are all 9 in*es or feet1 set ! m>k c 7 #aabh7 on ,c to diamet] on ,d & ov] l5g? on ,b1 read cubic cont5ts on ,a4 ,to f9d ! area ( an ellipse4 ,set #bje on ,c to #afa on ,d2 br+ cursor to l5g? ( major axis on ,c1 #a on ,c to cursor1 & "u l5g? ( m9or axis on ,c read area on ,d4 ,,ex4--,f9d ! area ( an ellipse ! major & m9or axes ( :i* are #af 94 & #ab 94 9 l5g? respectively4 ,set #bje on ,c to #afa on ,d2 br+ cursor to #af on ,c1 #a on ,c to cursor1 & "u #ab on ,c read #aejh 94 on ,d4 ,to f9d ! surface ( sph]es4 ,set #cadaf on ,b to ,r4,h4 or ,l4,h4 index ( ,a1 & ov] diamet] on ,d read by ! aid ( ! cursor1 ! convex surface on ,b4 ,to f9d ! cubic cont5ts ( sph]es4 ,set #aia on ,b to diamet] on ,a1 & ov] diamet] on ,c read cubic cont5ts on ,a4 ,,weights ,,( ,,metals4 ,to f9d ! weight 9 lb4 p] l9eal foot ( squ>e b>s ( metal4 ,set index ( ,b to weight ( #ab cubic in*es ( ! metal 7i4e41 "o l9eal foot1 #a squ>e in* 9 section7 on ,a1 & ov] ! side ( ! squ>e 9 in*es on ,c read weight 9 lb4 on ,a4 ,,ex4--,f9d ! weight p] foot l5g? ( #d;94 squ>e wr\ght-iron b>s4 ,set middle index ( ,b to #ccc on ,a1 & ov] #d on ,c read #fge lb4 on ,a4 7,n4,b4--,= o!r metals use ! correspond+ con/ant 9 column 7#b71 bel[74 ,to f9d ! weight 9 lb4 p] l9eal foot ( r\nd b>s4 ,set ,r4,h4 or ,l4,h4 index ( ,b to weight ( #ab cyl9drical in*es ( ! metal on ,a 7column 7#d71 bel[71 & opposite ! diamet] ( ! b> 9 in*es on ,c1 read weight 9 lb4 p] l9eal foot on ,a4 ,,ex4--,f9d ! weight ( #a l9eal foot ( #b 94 r\nd ca/ /eel4 ,set ,l4,h4 index ( ,b to #bfh on ,a1 & ov] #b on ,c read #ajg lb4 on ,a4 ,to f9d ! weight ( flat b>s 9 lb4 p] l9eal foot4 ,set ! bread? 9 in*es on ,c to 7#a7_/7weight ( #ab cub4 947 ( ! metal 7column 7#c71 bel[7 on ,d1 & above ! ?ickness on ,d read weight 9 lb4 p] l9eal foot on ,c4 ,,ex4--,f9d ! weight p] l9eal foot ( b> /eel1 #d;94 wide & 94 ?ick4 ,set #de on ,c to #jbid on ,d1 & ov] #jfbe on ,d read #ief lb4 p] l9eal foot on ,c4 ,to f9d ! weight p] squ>e foot ( %eet metal1 set ! weight p] cubic foot ( ! metal 7col4 #a7 on ,c to #ab on ,d1 & 7#a7 7#b7 7#c7 7#d7 ,weight 9 ,weight ( 7#a7_/7,wt4 ,weight ( ,metals4 lb4 p] #ab cubic ( #ab cub4 #ab cubic ft4 94 947 cyl9drical 94 ,wr\ght iron#dhj #ccc #jcjj #bfb ,ca/ iron #dej #cabe #jcbj #bde ,ca/ /eel #dij #cdj #jbid #bfh ,copp] #eej #chb #jbfb #cjj ,alum9ium #afh #aaff #jjhe #jiae ,brass #ebj #cfa #jbgg #bhc ,lead #gaj #dic #jbjc #chg ,t9 #dfb #cba #jcab #beb ,z9c 7ca/7 #dcj #bih #jcce #bcd 7%eet7 #dej #cabe #jcbj #bde above ! ?ickness ( ! plate 9 in*es on ,d read weight 9 lb4 p] squ>e foot on ,c4 ,,ex4--,f9d ! weight 9 lb4 p] squ>e foot ( alum9ium %eet 94 ?ick4 ,set #afh on ,c to #ab on ,d1 & ov] #jcge on ,d read #ebe lb4 on ,c4 ,to f9d ! weight ( pipes 9 lb4 p] l9eal foot4 ,set mean diamet] ( ! pipe 9 in*es 7i4e41 int]nal diamet] plus ! ?ickness1 or ext]nal diamet] m9us ! ?ickness7 on ,c to ! con/ant giv5 bel[ on ,d1 & ov] ! ?ickness on ,d read weight 9 lb4 p] l9eal foot on ,c4 ,metals4 ,con/ant = ,pipes4 ,con/ant = ,sph]es4 ,wr\ght iron #jjiee #fhg ,ca/ iron #jajbj #gce ,/eel #jjicf #fgc ,brass #jjhhb #fce ,copp] #jjhcd #fjj ,lead #jjfdf #dfe ,,ex4--,f9d ! weight p] foot ( ca/-iron pip+ #d 94 int]nal diamet] & 94 ?ick4 ,set #de on ,c to #jajb on ,d1 & ov] #je on ,d read #bba lb4 on ,c1 ! requir$ weight4 ,to f9d ! weight 9 lb4 ( sph]es or balls1 giv5 ! diamet] 9 in*es4 7,w #jebcf;d#c wt4 ( #a cub4 94 ( mat]ial74 ,set ! con/ant = sph]es 7giv5 above7 on ,b to diamet] 9 in*es on ,a1 & ov] diamet] on ,c read weight 9 lb4 on ,a4 ,,ex4--,f9d ! weight ( a ca/-iron ball #g;94 9 diamet]4 ,set #gce on ,b to #ge on ,a1 & ov] #ge on ,c read #egg lb4 on ,a4 ,to f9d diamet] 9 in*es ( a sph]e ( giv5 weight4 ,set ! cursor to ! giv5 weight 9 lb4 on ,a1 & move ! slide until ! same numb] is f\nd on ,c "u ! cursor t is simultane\sly f\nd on ,a ov] ! con/ant = ! sph]e on ,b4 ,,ex4--,f9d diamet] 9 in*es ( a sph]e ( ca/-iron to weigh #glb4 ,sett+ ! cursor to #ge on ,a1 & mov+ ! slide1 x is f\nd t :5 #ch on ,c falls "u ! cursor1 #ch on ,a is simultane\sly f\nd ov] #gce on ,b4 ,! requir$ diamet] is !re=e #ch 94 ,! rules = cubes & cube roots 7page #dj7 %\ld be kept 9 view 9 solv+ ! la/ two examples4 ,,fall+ ,,bodies4 ,to f9d velocity 9 feet p] second ( a fall+ body1 giv5 ! "t ( fall 9 seconds4 ,set index on ,c to "t ( fall on ,d1 & "u #cbb on ,c read velocity 9 feet p] second on ,d4 ,to f9d velocity 9 feet p] second1 giv5 di/ance fall5 ?r\gh 9 feet4 ,set #a on ,c to di/ance fall5 ?r\gh on ,a1 & "u #fdd on ,b read velocity 9 feet p] second on ,d4 ,,ex4--,f9d velocity acquir$ by fall+ ?r\gh #ad ft4 ,set 7,r4,h47 index ( ,c to #ad on ,a1 & "u #fdd on ,b read #cj ft4 p] second on ,d4 ,to f9d di/ance fall5 ?r\gh 9 feet 9 a giv5 "t4 ,set index ( ,c to "t 9 seconds on ,d1 & ov] #afa on ,b read di/ance fall5 ?r\gh 9 feet on ,a4 ,,c5trifugal ,,=ce4 ,to f9d ! c5trifugal =ce ( a revolv+ mass 9 lb4 ,set #bidj on ,b to revolutions p] m9ute on ,d2 br+ cursor to weight 9 lb4 on ,b2 index ( ,b to cursor1 & ov] radius 9 feet on ,b read c5trifugal =ce 9 lb4 on ,a4 ,to f9d ! c5trifugal /ress 9 lb4 p] squ>e in*1 9 rims ( revolv+ :eels ( ca/ iron4 ,set #fac on ,c to ! mean diamet] ( ! :eel 9 feet on ,d1 & ov] revolutions p] m9ute on ,c read /ress p] squ>e in* on ,a4 ,,ex4--,f9d ! /ress p] squ>e in* 9 a ca/-iron fly-:eel rim #h ft4 9 diamet] & runn+ at #abj revolutions p] m9ute4 ,set #fac on ,c to #h on ,d1 & ov] #abj on ,c read #bde lb4 p] squ>e in* on ,a4 ,,! ,,/eam ,,eng9e4 ,giv5 ! /roke & numb] ( revolutions p] m9ute1 to f9d ! pi/on spe$4 ,set /roke 9 in*es on ,c to #f on ,d1 & ov] numb] ( revolutions on ,d read pi/on spe$ 9 feet p] m9ute on ,c4 ,to f9d cubic feet ( /eam 9 a cyl9d] at cut-(f1 giv5 diamet] ( cyl9d] & p]iod ( admission 9 in*es4 ,set #bbjj on ,b to cyl9d] diamet] on ,d1 & ov] p]iod ( admission on ,b read cubic feet ( /eam on ,a4 ,,ex4--,cyl9d] diamet] #bf 941 /roke #dj 941 cut-(f at ( /roke4 ,f9d cubic feet ( /eam us$ 7!oretically7 p] /roke4 ,set #bbjj on ,b to #bf on ,d1 & ov] #dj or #be 94 on ,b1 read #gfh cub4 ft4 on ,a1 z ! numb] ( cubic feet ( /eam us$ p] /roke4 ,giv5 ! diamet] ( a cyl9d] 9 in*es1 & ! pressure 9 lb4 p] squ>e in*1 to f9d ! load on ! pi/on 9 tons4 ,set pressure 9 lb4 p] squ>e in* on ,b to #bheb on ,a1 & ov] cyl9d] diamet] 9 in*es on ,d read load on pi/on 9 tons on ,b4 ,,ex4--,/eam pressure #ahj lb4 p] squ>e in*2 cyl9d] diamet]1 #db 94 ,f9d load 9 tons on pi/on4 ,set #ahj on ,b to #bheb on ,a1 & ov] #db on ,d read #aaa tons1 ! gross load1 on ,b4 ,giv5 admission p]iod & absolute initial pressure ( /eam 9 a cyl9d]1 to f9d ! pressure at v>i\s po9ts 9 ! expansion p]iod 7iso!rmal expansion74 ,inv]t ! slide & set ! admission p]iod1 9 in*es1 on to ! initial pressure on ,d2 !n "u any po9t 9 ! expansion /roke on f9d ! correspond+ pressure on ,d4 ,,ex4--,admission p]iod #ab 941 /roke #db 941 initial pressure #hj lb4 p] squ>e in*4 ,f9d pressure at successive fif?s ( ! expansion p]iod4 ,set #ab on to #hj on ,d1 & opposite #ah1 #bd1 #cj1 #cf & #db 94 ( ! :ole /roke on f9d ! correspond+ pressures on ,d3--#ecc1 #dj1 #cb1 #bff & #bbh lb4 p] squ>e in*4 ,to f9d ! mean pressure con/ant = iso!rmally exp&+ /eam1 giv5 ! cut-(f z a fraction ( ! /roke4 ,f9d ! log>i?m ( ! ratio ( ! expansion r1 by ! me?od previ\sly expla9$ 7page #df74 ,prefix ! *>act]i/ic & to ! numb] ?us obta9$1 on ,d1 set #a on ,c4 ,!n "u #bcjb on ,c read x on ,d4 ,to x #a on ,d set r on ,c1 & "u index ( ,c read mean pressure con/ant on ,d4 ,! latt]1 multipli$ by ! initial pressure1 gives ! mean =w>d pressure ?r\gh\t ! /roke4 7,n4,b4--,common log4 #bcjb hyp]bolic log47 ,,ex4--,f9d ! mean pressure con/ant = a cut-(f ( ?1 or a ratio ( expansion ( #d4 ,set 7,l4,h47 index ( ,c to #d on ,d1 & on ! rev]se side ( ! slide read #jfjb on ! log>i?mic scale4 ,! *>act]i/ic #j2 h5ce to #jfjb on ,d set 7,r4,h47 index ( ,c1 & "u #bcjb on ,c read #achd on ,d4 ,add #a1 & to #bchd ?us obta9$ on ,d set r 7 #d7 on ,c1 & "u #a on ,c read #jeif1 ! mean pressure con/ant requir$4 ,mean pressure con/ants = ! mo/ usual degrees ( cut-(f are giv5 bel[3-- ,cut-(f 9 fractions ( /roke ,mean pressure con/ant #jifh #jieb #jicd #jiai #jiac #jhdf #jgff #jgej #jfii #jffd #jeif #jebb #jdfe #jdba #jche #jcee #jccj #jcji #jbij #jbgd #jbfj #jbdg #jbcf ,to f9d mean pressure3--,set #a on ,c to con/ant on ,d1 & "u initial pressure on ,c read mean pressure on ,d4 ,giv5 ! absolute initial pressure1 l5g? ( /roke1 & admission p]iod1 to f9d ! absolute pressure at any po9t 9 ! expansion p]iod1 x be+ assum$ t ! /eam exp&s adiabatically4 7,p#b 7,p#a7_/7,r7 9 :i* ,p#a initial pressure & ,p#b ! pressure correspond+ to a ratio ( expansion ,r47 ,set ,l4,h4 index ( ,c to ratio ( expansion on ,d1 & read on ! back ( ! slide ! decimal ( ! log>i?m4 ,add ! *>act]i/ic1 & to ! numb] ?us obta9$ on ,d set #i on ,c1 & read (f ! value f\nd on ,d "u ! index ( ,c4 ,set ?is numb] on ! log>i?mic scale to ! index m>k1 9 ! op5+ on ! back ( ! rule1 & "u ,l4,h4 index ( ,c read ! value ( ,r on ,d4 ,! initial pressure divid$ by ?is value gives ! correspond+ pressure due to ! expansion4 ,,ex4--,absolute initial pressure #abj lb4 p] squ>e in*2 /roke1 #d ft42 cut-(f 4 ,f9d ! respective pressures :5 & ?s ( ! /roke h be5 complet$4 ,9 ! fir/ case ,r #b4 ,!re=e sett+ ! ,l4,h4 index ( ,c to #b on ,d1 we f9d ! decimal ( ! log>i?m on ! back ( ! slide to be #jcja4 ,! *>act]i/ic is #j1 s plac+ #i on ,c to #jcja on ,d1 we read #jccd z ! value "u ! ,r4,h4 index ( ,c4 7,n4,b4--,9 locat+ ! decimal po9t x is to be obs]v$ t ! log4 ( ,r has be5 multipli$ by #aj1 9 accordance ) ! t]ms ( ! above expression47 ,sett+ ?is numb] on ! log>i?mic scale to ! back index1 ! value ( ,r is f\nd on ,d1 "u ! ,l4,h4 index ( ,c1 to be #baf4 ,sett+ #abj on ,c to ?is value1 x is f\nd t ! pressure at /roke1 read on ,c ov] ! ,r4,h4 index ( ,d1 is #eee lb4 p] squ>e in*4 ,9 a simil> mann]1 ! pressure :5 ?s ( ! /roke is complet$ is f\nd to be #ced lb4 p] squ>e in*4 ,= o!r conditions ( exp&+ /eam1 or = gas or air1 ! me?od ( proc$ure is simil> to ! above4 ,to f9d ! horse-p[] ( an eng9e1 hav+ giv5 ! mean effective pressure1 ! cyl9d] diamet]1 /roke1 & numb] ( revolutions p] m9ute4 ,to cyl9d] diamet] on ,d set #ade on ,c2 br+ cursor to /roke 9 feet on ,b1 #a on ,b to cursor1 cursor to numb] ( revolutions on ,b1 #a on ,b to cursor1 & ov] mean effective pressure on ,b f9d horse-p[] on ,a4 7,n4,b4--,if /roke is 9 in*es1 use #ejb 9 place ( #ade giv5 above47 ,,ex4--,f9d ! indicat$ horse-p[]1 giv5 cyl9d] diamet] #bg 941 mean effective pressure #ch lb4 p] squ>e in*1 /roke #cb 941 revolutions #eg p] m9ute4 ,set #ejb on ,c to #bg on ,d1 br+ cursor to #cb on ,b1 #a on ,b to cursor1 cursor to #eg on ,b1 #a on ,b to cursor1 & ov] #ch on ,b read #bjj ,i4,h4,p4 on ,a4 ,to det]m9e ! horse-p[] ( a comp\nd eng9e1 inv]t ! slide & set ! diamet] ( ! high-pressure cyl9d] on to ! cut-(f 9 t cyl9d] on ,a4 ,use ! numb] !n f\nd on ,a ov] ! diamet] ( ! l[-pressure cyl9d] on z ! cut-(f 9 t cyl9d]1 work+ ) ! same pressure & pi/on spe$1 & calculate ! horse-p[] z = a s+le cyl9d]4 ,to f9d ! cyl9d] ratio 9 comp\nd eng9es1 inv]t ! slide & set index ( to diamet] ( ! l[-pressure cyl9d] on ,d4 ,!n ov] ! diamet] ( ! high-pressure cyl9d] on ,c1 read cyl9d] ratio on ,a4 ,,ex4--,diamet] ( high-pressure cyl9d] #g;941 l[-pressure #ae 94 ,f9d cyl9d] ratio4 ,set index on to #ae on ,d1 & ov] #gge on read #cge1 ! requir$ ratio1 on ,a4 ,! cyl9d] ratios ( triple or quadruple-expansion eng9es may be simil>ly det]m9$4 ,,ex4--,9 a quadruple-expansion eng9e1 ! cyl9d]s are #ah1 #bf1 #cg1 & #ed in*es 9 diamet]4 ,f9d ! respective ratios ( ! high1 fir/ int]m$iate1 & second int]m$iate cyl9d]s to ! l[-pressure4 ,set 7,r4,h47 index ( to #ed on ,d1 & ov] #ah1 #bf1 & #cg on read #i1 #dca1 & #bac1 ! requir$ ratios1 on ,a4 ,giv5 ! mean effective pressures 9 lb4 p] squ>e in* 9 ea* ( ! ?ree cyl9d]s ( a triple-expansion eng9e1 ! ,i4,h4,p4 to be develop$ 9 ea* cyl9d]1 & ! pi/on spe$1 to f9d ! respective cyl9d] diamet]s4 ,set #db1jjj on ,b to pi/on spe$ on ,a2 br+ cursor to mean effective pressure 9 l[-pressure cyl9d] on ,b1 index ( ,b to cursor1 & "u ,i4,h4,p4 on ,a read l[-pressure cyl9d] diamet] on ,c4 ,to f9d ! diamet]s ( ! high-pressure & int]m$iate-pressure cyl9d]s1 inv]t ! slide & place ! mean pressure 9 ! l[-pressure cyl9d] on to ! diamet] ( t cyl9d] on ,d4 ,!n "u ! respective mean pressures on read correspond+ cyl9d] diamet]s on ,d4 ,,ex4--,! mean effective pressures 9 ! cyl9d]s ( a triple-expansion eng9e are3--,l4,p41 #ajcb2 ,i4,m4,p41 #bge2 & ,h4,p41 #gge lb4 p] squ>e in*4 ,! pi/on spe$ is #fej ft4 p] m9ute1 & ! ,i4,h4,p4 develop$ 9 ea* cyl9d]1 #gej4 ,f9d ! cyl9d] diamet]s4 ,set #db1jjj on ,b to #fej on ,a1 & br+ cursor to #ajcb on ,b4 ,br+ index ( ,b to cursor1 & "u #gej on ,a read #fhe 94 on ,c1 ! ,l4,p4 cyl9d] diamet]4 ,inv]t ! slide1 & plac+ #ajcb on to #fhe on ,d1 read1 "u #bge on 1 ! ,i4,m4,p4 cyl9d] diamet] #db 941 on ,d2 also "u #gge on read ! ,h4,p4 cyl9d] diamet] #be 941 on ,d4 ,to compute brake or dynamometrical horse-p[]4 ,set #ebe on ,c to ! total weight 9 lb4 act+ at ! end ( ! lev] 7or pull ( spr+ balance 9 lb47 on ,d2 set cursor to l5g? ( lev] 9 feet on ,c1 br+ #a on ,c to cursor1 & "u numb] ( revolutions p] m9ute on ,c f9d brake horse-p[] on ,d4 ,giv5 cyl9d] diamet] & pi/on spe$ 9 feet p] m9ute1 to f9d diamet] ( /eam pipe1 assum+ ! maximum velocity ( ! /eam to be #fjjj ft4 p] m9ute4 ,set #fjjj on ,b to cyl9d] diamet] on ,d1 & "u pi/on spe$ on ,b read /eam pipe diamet] on ,d4 ,giv5 ! numb] ( revolutions p] m9ute ( a ,watt gov]nor1 to f9d ! v]tical height 9 in*es1 f ! plane ( revolution ( ! balls to ! po9t ( susp5sion4 ,set revolutions p] m9ute on ,c to #ce1bjj on ,a1 & ov] index ( ,b read height on ,a4 ,giv5 ! weight 9 lb4 ( ! rim ( a ca/-iron fly-:eel1 to f9d ! sectional area ( ! rim 9 squ>e in*es4 ,set ! mean diamet] ( ! :eel 9 feet on ,c to #jajb on ,d1 & "u weight ( rim on ,c f9d area on ,d4 ,giv5 ! consumption ( coal 9 tons p] week ( #ef h\rs1 & ! ,i4,h4,p41 to f9d ! coal consum$ p] ,i4,h4,p4 p] h\r4 ,set ,i4,h4,p4 on ,c to #dj on ,d1 & "u weekly consumption on ,c read lb4 ( coal p] ,i4,h4,p41 p] h\r on ,d4 ,,ex4--,f9d coal us$ p] ,i4,h4,p4 p] h\r1 :5 #bd tons is ! weekly consumption = #cjj ,i4,h4,p4 ,set #cjj on ,c to #dj on ,d1 & "u #bd on ,c read #cb lb4 p] ,i4,h4,p4 p] h\r on ,d4 7,n4,b4--,= any o!r numb] ( work+ h\rs p] week divide #bbdj by ! numb] ( work+ h\rs1 & use ! quoti5t 9 place ( #dj z above47 ,to f9d ! tractive =ce ( a locomotive4 ,set diamet] ( driv+ :eel 9 in*es on ,b to diamet] ( cyl9d] 9 in*es on ,d1 & ov] ! /roke 9 in*es on ,b read on ,a1 tractive =ce 9 lb4 = ea* lb4 ( effective pressure on ! pi/on4 ,,/eam ,,boil]s4 ,to f9d ! bur/+ pressure ( a cyl9drical boil] %ell1 hav+ giv5 ! diamet] ( %ell & ! ?ickness & ultimate /r5g? ( ! mat]ial4 ,set ! diamet] ( ! %ell 9 in*es on ,c to twice ! ?ickness ( ! plate on ,d1 & "u /r5g? ( mat]ial p] squ>e in* on ,c read bur/+ pressure 9 lb4 p] squ>e in* on ,d4 ,,ex4--,f9d ! bur/+ pressure ( a cyl9drical boil] %ell #g ft4 #f 94 9 diamet]1 ) plates 94 ?ick1 assum+ an ultimate /r5g? ( #ej1jjj lb4 p] squ>e in*4 ,set #ij on ,c to #aj on ,d1 & "u #ej1jjj on ,c f9d #eee lb4 on ,d4 ,to f9d work+ pressure = ,fox's corrugat$ furnaces by ,bo>d ( ,trade rule4 ,set ! lea/ outside diamet] 9 in*es on ,c to #ad1jjj on ,d1 & "u ?ickness 9 in*es on ,c read work+ pressure on ,d 9 lb4 p] squ>e in*4 ,to f9d diamet] d 9 in*es1 ( r\nd /eel = safety valve spr+s by ,bo>d ( ,trade rule4 ,set #hjjj on ,c to load on spr+ 9 lb4 on ,d1 & "u ! mean diamet] ( ! spr+ 9 in*es on ,c read d#c on ,d4 ,!n extract ! cube root z p] rule4 ,,spe$ ,,ratios ,,( ,,pulleys1 ,,etc4 ,giv5 ! diamet] ( a pulley & its numb] ( revolutions p] m9ute1 to f9d ! circumf]5tial velocity ( ! pulley or ! spe$ ( ropes1 belts1 etc41 driv5 !reby4 ,set diamet] ( pulley 9 in*es on ,c to #chb on ,d1 & ov] revolutions p] m9ute on ,d read spe$ 9 feet p] m9ute on ,c4 ,,ex4--,f9d ! spe$ ( a belt driv5 by a pulley #ec 94 9 diamet] & runn+ at #ahj revolutions p] m9ute4 ,set #ec on ,c to #chb on ,d1 & ov] #ahj on ,d read #bejj ft4 p] m9ute on ,c4 ,,ex4--,f9d ! spe$ ( ! pit* l9e ( a spur :eel #c ft4 #f 94 9 diamet] runn+ at #fj revolutions p] m9ute4 ,set #db 94 on ,c to #chb on ,d1 & ov] #fj on ,d read #ffj ft4 p] m9ute on ,c4 ,giv5 diamet] & numb] ( revolutions p] m9ute ( a driv+ pulley1 & ! diamet] ( ! driv5 pulley1 to f9d ! numb] ( revolutions ( ! latt]4 ,inv]t ! slide & set diamet] ( driv+ pulley on to giv5 numb] ( its revolutions on ,d2 !n opposite diamet] ( any driv5 pulley on read its numb] ( revolutions on ,d4 ,,ex4--,diamet] ( driv+ pulley #aj ft42 revolutions p] m9ute #ee2 diamet] ( driv5 pulley #b ft4 #i 94 ,f9d numb] ( revolutions p] m9ute ( latt]4 ,set #aj on to #ee on ,d1 & opposite #bge on read #bjj revolutions on ,d4 ,,belts ,,& ,,ropes4 ,to f9d ! ratio ( t5sions 9 ! two sides ( a belt1 giv5 ! coeffici5t ( friction betwe5 belt & pulley & ! numb] ( degrees 9 ! arc ( contact 7log4 ,r 77_/7#acb774 ,set #acb on ,c to ! coeffici5t ( friction on ,d1 & read (f ! value f\nd on ,d "u ! numb] ( degrees 9 ! arc ( contact on ,c4 ,place ?is value on ! scale ( equal p>ts on ! back ( ! slide1 to ! index m>k 9 ! ap]ture1 & read ! requir$ ratio on ,d "u ! ,l4,h4 index ( ,c4 ,,ex4--,f9d ! t5sion ratio 9 a belt1 assum+ a coeffici5t ( friction ( #jc & an arc ( contact ( #abj degrees4 ,set #acb on ,c to #jc on ,d1 & "u #abj on ,c read #jbgc4 ,place ?is on ! scale to ! index on ! back ( ! rule1 & "u ! ,l4,h4 index ,c read #ahge on ,d1 ! requir$ ratio4 ,giv5 belt velocity & horse-p[] to be transmitt$1 to f9d ! requisite wid? ( belt1 tak+ ! effective t5sion at #ej lb4 p] in* ( wid?4 ,set #ffj on ,c to velocity 9 feet p] m9ute on ,d1 & opposite horse-p[] on ,d f9d wid? ( belt 9 in*es on ,c4 ,giv5 velocity & wid? ( belt1 to f9d horse-p[] transmitt$4 ,set #ffj on ,c to velocity on ,d1 & "u wid? on ,c f9d horse-p[] transmitt$ on ,d4 7,n4,b4--,= any o!r effective t5sion1 in/ead ( #ffj use z a gauge po9t3--#cc1jjj t5sion47 ,giv5 spe$ & diamet] ( a cotton driv+ rope1 to f9d p[] transmitt$1 disreg>d+ c5trifugal action1 & assum+ an effective work+ t5sion ( #bjj lb4 p] squ>e in* ( rope4 ,set #baj on ,b to #age on ,d1 & ov] spe$ 9 feet p] m9ute on ,b read horse-p[] on ,a4 ,,ex4--,f9d ! p[] transmitt$ by a #a;94 rope runn+ at #djjj ft4 p] m9ute4 ,set #baj on ,b to #age on ,d1 & ov] #djjj on ,b read #ehc horse-p[] on ,a4 ,f9d ! 8c5trifugal t5sion0 9 ! previ\s example1 tak+ ! weight p] foot ( ! rope z #jbg;d#b4 ,set #fee on ,c to ! diamet]1 #age 941 on ,d1 & ov] ! spe$1 #djjj ft4 on ,c1 read c5trifugal t5sion #aad lb4 on ,a4 ,,spur ,,:eels4 ,giv5 diamet] & pit* ( a spur :eel1 to f9d numb] ( tee?4 ,set pit* on ,c to 7#cadaf7 on ,d1 & "u any diamet] on ,c read numb] ( tee? on ,d4 ,giv5 diamet] & numb] ( tee? 9 a spur :eel1 to f9d ! pit*4 ,set diamet] on ,c to numb] ( tee? on ,d1 & read pit* on ,c opposite #cadaf on ,d4 ,giv5 ! di/ance betwe5 ! c5tres ( a pair ( spur :eels & ! numb] ( revolutions ( ea*1 to det]m9e _! diamet]s4 ,to twice ! di/ance betwe5 ! c5tres on ,d1 set ! sum ( ! numb] ( revolutions on ,c1 & "u ! revolutions ( ea* :eel on ,c f9d ! respective :eel diamet]s on ,d4 ,,ex4--,! di/ance betwe5 ! c5tres ( two spur :eels is #cge 941 & !y are requir$ to make #ba & #bd revolutions 9 ! same "t4 ,f9d _! respective diamet]s4 ,set #ba #bd #de on ,c to #ge 7or #cge #b7 on ,d1 & "u #ba & #bd on ,c f9d #ce & #dj 94 on ,d z ! respective diamet]s4 ,to f9d ! p[] transmitt$ by too!d :eels1 giv5 ! pit* diamet] d 9 in*es1 ! numb] ( revolutions p] m9ute n1 & ! pit* p 9 in*es1 by ! rule1 ,h4,p4 7n d p#b7_/7djj74 ,set #djj on ,b to pit* 9 in*es on ,d2 set cursor to d on ,b1 #a on ,b to cursor1 & ov] any numb] ( revolutions n on ,b read p[] transmitt$ on ,a4 ,,ex4--,f9d ! horse-p[] capable ( be+ transmitt$ by a spur :eel #g ft4 9 diamet]1 #c 94 pit*1 & runn+ at #ij revolutions p] m9ute4 ,set #djj on ,b to #c on ,d2 br+ cursor to #hd 94 on ,b1 #a on ,b to cursor1 & ov] #ij revolutions on ,b read #agj1 ! horse-p[] transmitt$1 on ,a4 ,,screw-,,cutt+4 ,giv5 ! numb] ( ?reads p] in* 9 ! guide screw1 to f9d ! :eels to cut a screw ( giv5 pit*4 ,set ?reads p] in* 9 guide screw on ,c1 to ! numb] ( ?reads p] in* to be cut on ,d4 ,!n opposite any numb] ( tee? 9 ! :eel on ! m&rel on ,c1 is ! numb] ( tee? 9 ! :eel to be plac$ on ! guide screw on ,d4 ,,/r5g? ,,( ,,%aft+4 ,giv5 ! diamet] d ( a /eel %aft1 & ! numb] ( revolutions p] m9ute n1 to f9d ! horse-p[] f3-- ,h4,p4 d#c n #jjb4 ,set #a on ,c to d on ,d1 & br+ cursor to d on ,b4 ,br+ #ej on ,b to cursor1 & ov] numb] ( revolutions on ,b read ,h4,p4 on ,a4 ,,ex4--,f9d horse-p[] transmitt$ by a #c 94 /eel %aft at #aaj revolutions p] m9ute4 ,set #a on ,c to #c on ,d1 & br+ cursor to #c on ,b4 ,br+ #ej on ,b to cursor1 & ov] #aaj on ,b read #eid horse-p[] on ,a4 ,giv5 ! horse-p[] to be transmitt$ & ! numb] ( revolutions ( a /eel %aft1 to f9d ! diamet]4 ,set revolutions on ,b to horse-p[] on ,a1 & br+ cursor to #ej on ,b4 ,!n move ! slide until ! same numb] is f\nd on ,b "u ! cursor t is simultane\sly f\nd on ,d "u ! index ( ,c4 ,?is numb] is ! diamet] requir$4 ,to f9d ! deflection k 9 in*es1 ( a r\nd /eel %aft ( diamet] d1 "u a uni=mly di/ribut$ load 9 lb4 w1 & support$ by be>+s1 ! c5tres ( :i* are l feet ap>t 7k 7w l#c7_/7gh1jjj;d#d774 ,modify+ ! =m ( ?is expression slightly1 we proce$ z foll[s3--,set d on ,c to l on ,d1 & br+ ! cursor to ! same numb] on ,b t is f\nd on ,d "u ! index ( ,c4 ,br+ d on ,b to cursor1 cursor to w on ,b1 #gh1hjj on ,b to cursor1 & read deflection on ,a ov] index ( ,b4 ,,ex4--,f9d ! deflection 9 in*es ( a r\nd /eel %aft #c;94 diamet]1 c>ry+ a uni=mly di/ribut$ load ( #cbjj lb41 ! di/ance ap>t ( ! c5tres ( support be+ #i ft4 ,set #ce on ,c to #i on ,d1 & read #beg on ,d1 "u ! ,l4,h4 index ( ,c4 ,set cursor to #beg on ,b1 & br+ #ce on ,b to cursor1 cursor to #cbjj on ,b1 #gh1jjj on ,b to cursor1 & ov] ,l4,h4 index ( ,b read #jaii 941 ! requir$ deflection on ,a4 ,to f9d ! diamet] ( a %aft subject to twi/+ only1 giv5 ! twi/+ mom5t 9 in*-lb4 & ! all[able /ress 9 lb4 p] squ>e in*4 ,set ! /ress 9 lb4 p] squ>e in* on ,b to ! twi/+ mom5t 9 in*-lb4 on ,a1 & br+ cursor to #ea on ,b4 ,!n move ! slide until ! same numb] is f\nd on ,b "u ! cursor t is simultane\sly f\nd on ,d "u ! index ( ,c4 ,,ex4--,a /eel %aft is subject$ to a twi/+ mom5t ( #b1gjj1jjj in*-lb4 ,det]m9e ! diamet] if ! all[able /ress is tak5 at #ijjj lb4 p] squ>e in*4 ,set #ijjj on ,b to #b1gjj1jjj on ,a1 & br+ ! cursor to #ea on ,b4 ,mov+ ! slide to ! left1 x is f\nd t :5 #aaea on ! ,r4,h4 scale ( ,b is "u ! cursor1 ! ,l4,h4 index ( ,c is opposite #aaea on ,d4 ,?is1 !n1 is ! requir$ diamet] ( ! %aft4 7,n4,b4--,! rules = ! scales to be us$ 9 f9d+ ! cube root 7page #db7 mu/ be c>efully obs]v$ 9 work+ ^! examples47 ,,mom5ts ,,( ,,in]tia4 ,to f9d ! mom5t ( in]tia ( a squ>e section ab\t an axis =m$ by "o ( its diagonals 7,i 7s#d7_/7ab774 ,set index ( ,c to ! l5g? ( ! side ( squ>e s on ,d2 br+ cursor to s on ,c1 #ab on ,b to cursor1 & ov] index ( ,b read mom5t ( in]tia on ,a4 ,to f9d ! mom5t ( in]tia ( a rectangul> section ab\t an axis p>allel to "o side & p]p5dicul> to ! plane ( b5d+4 ,set index ( ,c to ! height or dep? h ( ! section1 & br+ cursor to h on ,b4 ,set #ab on ,b to cursor1 & ov] bread? b ( ! section on ,b read mom5t ( in]tia on ,a4 ,,ex4--,f9d ! mom5t ( in]tia ( a rectangul> section ( :i* h #ad 94 & b #g 94 ,set index ( ,c to #ad on ,d1 & cursor to #ad on ,b4 ,br+ #ab on ,b to cursor1 & ov] #g on ,b read #afjj on ,a4 ,,dis*>ge ,,f ,,pumps1 ,,pipes1 ,,etc4 ,to f9d ! !oretical deliv]y ( pumps1 9 gallons p] /roke4 ,set #bid on ,b to ! diamet] ( ! plung] 9 in*es on ,d1 & ov] l5g? ( /roke 9 feet on ,b read !oretical deliv]y 9 gallons p] /roke on ,a4 7,n4,b4--,a d$uction ( f #bj to #dj p] c5t4 %\ld be made to all[ = slip47 ,to f9d loss ( head ( wat] 9 feet due to friction 9 pipes 7,prony's rule74 ,set diamet] ( pipe 9 feet on ,b to velocity ( wat] 9 feet p] second on ,d & br+ cursor to #bbe on ,b2 br+ #a on ,b to cursor1 & ov] l5g? ( pipe 9 miles on ,b1 read loss ( head ( wat] 9 feet1 on ,a4 ,to f9d velocity 9 feet p] second1 ( wat] 9 pipes 7,blackwell's rule74 ,set #bc on ,b to diamet] ( pipe 9 feet on ,a1 & "u incl9ation ( pipe 9 feet p] mile on ,b read velocity 9 feet p] second on ,d4 ,to f9d ! dis*>ge ov] weirs 9 cubic feet p] m9ute & p] foot ( wid?4 7,dis*>ge #bad7;h#c77 ,set #jjjdfg on ,c to ! head 9 feet h on ,d1 & "u h on ,b read dis*>ge on ,d4 ,to f9d ! !oretical velocity ( wat] fl[+ "u a giv5 head 9 feet4 ,set index ( ,b to head 9 feet on ,a1 & "u #fdd on ,b read !oretical velocity 9 feet p] second on ,d4 ,,horse-,,p[] ,,( ,,wat] ,,:eels4 ,to f9d ! effective horse-p[] ( a ,poncelet wat] :eel4 ,set #hhj on ,c to cubic feet ( fl[ ( wat] p] m9ute on ,d1 & "u height ( fall 9 feet on ,c1 read effective horse-p[] on ,d4 ,= brea/ wat] :eels use #ifj1 & = ov]%ot :eels #gge1 9 place ( #hhj z above4 ,,electrical ,,eng9e]+4 ,to f9d ! resi/ance p] mile1 9 ohms1 ( copp] wire ( high conductivity1 at #fj ,f4 ! diamet] be+ giv5 9 mils4 7#a mil4 #jjja 9474 ,set diamet] ( wire 9 mils4 on ,c to #ed1ijj on ,a1 & ov] ,r4,h4 or ,l4,h4 index ( ,b read resi/ance 9 ohms on ,a4 ,,ex4--,f9d ! resi/ance p] mile ( a copp] wire #fd mils4 9 diamet]4 ,set #fd on ,c to #ed1ijj on ,a1 & ov] ,r4,h4 index ( ,b read #acd ohms on ,a4 ,to f9d ! weight ( copp] wire 9 lb4 p] mile4 ,set #gia on ,c to diamet] ( wire 9 mils4 on ,d1 & ov] index ( ,b read weight p] mile on ,a4 ,giv5 electromotive =ce & curr5t1 to f9d electrical horse-p[]4 ,set #gdf on ,c to electromotive =ce 9 volts on ,d1 & "u curr5t 9 ampres on ,c read electrical horse-p[] on ,d4 ,giv5 ! resi/ance ( a circuit 9 ohms & curr5t 9 ampres1 to f9d ! en]gy absorb$ 9 horse-p[]4 ,set #gdf on ,b to curr5t on ,d1 & ov] resi/ance on ,b read en]gy absorb$ 9 ,h4,p4 on ,a4 ,,ex4--,f9d ! ,h4,p4 exp5d$ 9 s5d+ a curr5t ( #ae ampres ?r\gh a circuit ( #bbj ohms resi/ance4 ,set #gdf on ,b to #ae on ,d1 & ov] #bbj on ,b read #ffc ,h4,p4 on ,a4 ,,comm]cial4 ,to add on p]c5tages4 ,set #ajj on ,c to #ajj giv5 p]c5tage on ,d1 & "u orig9al numb] on ,c read result on ,d4 ,to d$uct p]c5tages4 ,set ,r4,h4 index ( ,c to #ajj ! giv5 p]c5tage on ,d1 & "u orig9al numb] on ,c read result on ,d4 ,,ex4--,f #af d$uct #g p] c5t4 ,set #aj on ,c to #ibe on ,d & "u #af on ,c1 read #adh #ad1 #afs4 on ,d4 ,to calculate simple int]e/4 ,set #a on ,c to rate p] c5t4 on ,d2 br+ cursor to p]iod on ,c & #a on ,c to cursor4 ,!n opposite any sum on ,c f9d simple int]e/ on ,d4 ,= int]e/ p] annum4 ,set ,r4,h4 index on ,c to rate on ,d1 & opposite pr9cipal on ,c read int]e/ on ,d4 ,,ex4--,f9d ! am\nt ) simple int]e/ ( #bej at #h p] c5t41 & = a p]iod ( #a ye> & #i mon?s4 ,set #a on ,c to #h on ,d2 br+ cursor to #age on ,c1 & #a on ,c to cursor2 !n opposite #bej on ,c read #ce1 ! int]e/1 on ,d4 ,!n #bej #ce #bhe ! am\nt4 ,to calculate comp\nd int]e/4 ,set ! ,l4,h4 index ( ,c to ! am\nt ( #a at ! giv5 rate ( int]e/ on ,d1 & f9d ! log>i?m ( ?is by read+ on ! rev]se side ( ! rule1 z expla9$ on page #df4 ,multiply ! log>i?m1 s f\nd1 by ! p]iod1 & set ! result1 on ! scale ( equal p>ts1 to ! index on ! "u-side ( ! rule2 !n opposite any sum on ,c read ! am\nt 7includ+ comp\nd int]e/7 on ,d4 ,,ex4--,f9d ! am\nt ( #ejj at #e p] c5t4 = #f ye>s1 ) comp\nd int]e/4 ,set ,l4,h4 index ( ,c to #aje on ,d1 & read at ! index on ! scale ( equal p>ts on ! "u-side ( rule1 #jjbab4 ,multiply by #f1 we obta9 #jabgb1 :i*1 on ! scale ( equal p>ts1 is plac$ to ! index 9 ! not* at ! end ( ! rule4 ,!n opposite #ejj on ,c read #fgj on ,d1 ! am\nt requir$1 includ+ comp\nd int]e/4 ,,miscellane\s ,,calculations4 ,to calculate p]c5tages ( compositions4 ,set weight 7or volume7 ( sample on ,c1 to weight 7or volume7 ( sub/ance consid]$1 on ,d2 !n "u index ( ,c read requir$ p]c5tage on ,d4 ,,ex4--,a sample ( coal weigh+ #abe grms4 conta9s #jjddbe grm4 ( a%4 ,f9d ! p]c5tage ( a%4 ,set #abe on ,c to #jjddbe on ,d1 & "u index on ,c read #ced1 ! requir$ p]c5tage ( a% on ,d4 ,giv5 ! /eam pressure ,p & ! diamet] d 9 millimetres1 ( ! ?roat ( an injector1 to f9d ! weight ,w1 ( wat] deliv]$ 9 lb4 p] h\r f ,w 7d#b,p7_/7#jeje74 ,set #jeje on ,c to ,p on ,a2 br+ cursor to d on ,c & index ( ,c to cursor4 ,!n "u d on ,c read deliv]y ( wat] on ,d4 ,to f9d ! pressure ( w9d p] squ>e foot1 due to a giv5 velocity 9 miles p] h\r4 ,set #a on ,b to #b on ,a1 & ov] ! velocity 9 miles p] h\r on ,d read pressure 9 lb4 p] squ>e foot on ,b4 ,to f9d ! k9etic en]gy ( a mov+ body4 ,set #fdd on ,b to velocity 9 feet p] second on ,d1 & ov] weight ( body 9 lb4 on ,b read k9etic en]gy or accumulat$ "w 9 foot-lb4 on ,a4 ,,trigonometrical ,,applications ,scales4--,n ! lea/ important feature ( ! mod]n slide rule is ! provision ( ! special scales on ! "u-side ( ! slide1 & by ! use ( :i*1 9 conjunction ) ! ord9>y scales on ! rule1 a l>ge v>iety ( trigonometrical computations may be readily p]=m$4 ,?ree scales w be f\nd on ! rev]se or "u-side ( ! slide ( ! ord9>y ,gravt or ,mannheim rule4 ,"o ( ^! is ! ev5ly-divid$ scale or scale ( equal p>ts ref]r$ to 9 previ\s sections1 & by :i*1 z expla9$1 ! decimal p>ts or mantiss ( log>i?ms ( numb]s may be obta9$4 ,usually ?is scale is ! c5tre "o ( ! ?ree1 b 9 "s rules x w be f\nd occupy+ ! l[e/ position1 9 :i* case "s little modification ( ! foll[+ in/ructions w be necess>y4 ,! requisite transpositions w1 h[ev]1 be evid5t :5 ! purposes ( ! scales are und]/ood4 ,! upp] ( ! ?ree scales1 usually di/+ui%$ by ! lett] ,s1 is a scale giv+ ! log>i?ms ( ! s9es ( angles1 & is us$ to det]m9e ! natural s9es ( angles ( f #ce m9utes to #ij degrees4 ,! notation ( ?is scale w be evid5t on inspection4 ,! ma9 divisions #a1 #b1 #c1 etc41 repres5t ! degrees ( angles2 b ! values ( ! subdivisions diff] accord+ to _! position on ! scale4 ,?us1 if any prim>y space is subdivid$ into #ab p>ts1 ea* ( ! latt] w be read z #e m9utes 7#e71 s9ce #a #fj4 ,s9es ( ,angles4--,to f9d ! s9e ( an angle ! slide is plac$ 9 ! groove1 ) ! "u-side upp]mo/1 & ! end division l9es or indices on ! slide1 co9cid+ ) ! "r & left indices ( ! ,a scale4 ,!n ov] ! giv5 angle on ,s is read ! value ( ! s9e ( ! angle on ,a4 ,if ! result is f\nd on ! left scale ( ,a 7#a to #aj71 ! log>i?mic *>act]i/ic is #b2 if x is f\nd on ! "r-h& side 7#aj to #ajj71 x is #a4 ,9 o!r words1 results on ! "r-h& scale are prefix$ by ! decimal po9t only1 :ile ^? on ! left-h& scale are to be prec$$ by a cyph] also4 ,?us3-- ,s9e #b #dj #jjdfe2 s9e #ae #dj #jbgj4 ,multiplication & division ( ! s9es ( angles are p]=m$ 9 ! same mann] z ord9>y calculations1 except+ t ! slide has its "u-face plac$ upp]mo/1 z j expla9$4 ,?us to multiply s9e #ae #dj by #ae1 ! ,r4,h4 index ( ,s is br\ght to #ae on ,a1 & opposite #ae #dj on ,s is f\nd #dje on ,a4 ,aga91 to divide #adb by s9e #af #cj1 we place #af #cj on ,s to #adb on ,a1 & ov] ,r4,h4 index ( ,s read #ejj on ,a4 ,! rules = ! numb] ( integ]s 9 ! results are ?us det]m9$3 ,let ,n be ! numb] ( integ]s 9 ! multipli] ,m or 9 ! divid5d ,d4 ,!n ! numb] ( integ]s ,p1 9 ! product or ,q1 9 ! quoti5t are z foll[s3-- ,:5 ! result is f\nd to ! "r ( ,m or ,d1 ,p ,n #b,q ,n & 9 ! same scale ,:5 ! result is f\nd to ! "r ( ,m or ,d1 ,p ,n #a,q ,n #a & 9 ! o!r scale ,:5 ! result is f\nd to ! left ( ,m or ,d1 &,p ,n #a,q ,n #a 9 ! o!r scale ,:5 ! result is f\nd to ! left ( ,m or ,d1 &,p ,n ,q ,n #b 9 ! same scale ,if ! division is ( ! =m 7#bj #cj7_/7ej71 ! result _c be read (f directly on ! face ( ! rule4 ,?us1 if 9 ! above example #bj #cj on ,s1 is plac$ to agree ) #ej on ! "r-h& scale ( ,a1 ! result f\nd on ,s "u ! ,r4,h4 index ( ,a is #dd #cj4 ,! requir$ num]ical value c !n be f\nd3 7#a7 ,by plac+ ! slide ) all indices co9cid5t :5 opposite #dd #cj on ,s w be f\nd #jjjg on ,a2 or 7#b7 ,9 ! ord9>y =m ( rule1 by read+ (f on ! scale ,b opposite ! index m>k 9 ! op5+ on ! "u-side ( ! rule4 ,! above rules = ! numb] ( integ]s 9 ! quoti5t d n apply 9 ?is case4 ,if x is requir$ to f9d ! s9e ( an angle simply1 ?is may be done ) ! slide 9 its ord9>y position1 ) scale ,b "u ,a4 ,! giv5 angle on scale ,s is !n set to ! index on ! "u-side ( ! rule1 & ! value ( ! s9e is read (f on ,b "u ! "r index ( ,a4 ,ow+ to ! rapidly dim9i%+ diff]5ces ( ! values ( ! s9es z ! upp] end ( ! scale is approa*$1 ! s9es ( angles betwe5 #fj & #ij _c be accurately det]m9$ 9 ! =ego+ mann]4 ,x is !re=e advisable to calculate ! value ( ! s9e by means ( ! =mula3 ,s9e #a #b s9#b 7#ij 7_/7#b74 ,to det]m9e ! value ( s9#b 7#ij 7_/7#b74 ,) ! slide 9 ! normal position1 set ! value ( 7#ij 7_/7#b74 on ,s to ! index on ! "u-side ( ! rule1 & read (f ! value x on ,b "u ! ,r4,h4 index ( ,a4 ,)\t mov+ ! slide f9d x on ,a1 & read "u x on ,b ! value requir$4 ,,ex4--,f9d value ( s9e #gi #dj4 ,s9e #gi #dj #a #bs9#b #e #aj4 ,b s9e #e #aj #jjijj1 & "u ?is value on ,a is #jjjha on ,b4 ,!re=e s9e #gi #dj #a #jjafb #jihch4 ,! s9es ( v small angles1 be+ v ne>ly proportional to ! angles !mselves1 are f\nd by direct read+4 ,to facilitate ?is1 "s rules are provid$ ) two m>ks1 "o ( :i*1 a s+le acc5t 771 corresponds to ! log>i?m ( 7#a7_/7s9e #a7 & is f\nd at ! numb] #cdch4 ,! o!r m>k--a d\ble acc5t 77--corresponds to ! log>i?m ( 7#a7_/7s9e #a7 & is f\nd at ! numb] #bjf1bfe4 ,9 "s rules ^! m>ks are f\nd on ei!r ! ,a or ! ,b scales2 sometimes !y are on bo?4 ,9 ei!r case ! angle on ! "o scale is plac$ s z to co9cide ) ! significant m>k on ! o!r1 & ! result read (f on ! fir/-nam$ scale opposite ! index ( ! second4 ,9 s9es ( angles "u #c1 ! numb] ( integ]s 9 ! result is #e2 :ile x is #d = angles f #c to #ba2 #c f #ba to #c #bg2 & #b f #c #bg to #cd #bc4 ,,ex4--,f9d s9e #f4 ,plac+ ! significant m>k = m9utes co9cid5t ) #f1 ! value opposite ! index is f\nd to be #age1 & by ! rule above ?is is to be read #jjjage4 ,= angles 9 seconds ! o!r significant m>k is us$2 :ile angles express$ 9 m9utes & seconds are to be fir/ r$uc$ to seconds4 ,?us1 #c #aj #aij4 ,tang5ts ( ,angles4--,!re rema9s to be consid]$ ! ?ird scale f\nd on ! back ( ! slide1 & usually di/+ui%$ f ! o!rs by be+ lett]$ ,t4 ,9 mo/ ( ! m rec5t =ms ( rule ?is scale is plac$ ne> ! l[] edge ( ! slide1 b 9 "s arrangem5ts x is f\nd to be ! c5tre scale ( ! ?ree4 ,aga91 9 "s rules ?is scale is figur$ 9 ! same direction z ! scale ( s9es--viz41 f left to "r1--:ile 9 o!rs ! ,t scale is rev]s$4 ,9 bo? cases !re is n[ usually an ap]ture =m$ 9 ! back ( ! left extremity ( ! rule1 ) an index m>k simil> to t already ref]r$ to 9 connection ) ! scale ( s9es4 ,consid]+ :at has be5 ref]r$ to z ! m g5]al arrangem5t1 ! me?od ( det]m9+ ! tang5ts ( angles may be ?us expla9$3-- ,! tang5t scale w be f\nd to comm5ce1 9 "s rules1 at ab\t #cd1 or1 precisely1 at ! angle ^: tang5t is #jja4 ,m usually1 h[ev]1 ! scale w be f\nd to comm5ce at ab\t #e #dc1 or at ! angle ^: tang5t is #ja4 ,! o!r extremity ( ! scale corresponds 9 all cases to #de1 or ! angle ^: tang5t is #a4 ,?is explanation w sugge/ ! me?od ( us+ ! scale1 h[ev] x may be arrang$4 ,if ! graduations comm5ce ) #cd1 ! ,t scale is to be us$ 9 conjunction ) ! "r & left scales ( ,a2 :ile if !y comm5ce ) #e #dc x is to be us$ 9 conjunction ) ! ,d scale4 ,9 ! =m] case ! slide is to be plac$ 9 ! rule s t ! ,t scale is adjac5t to ! ,a scales1 &1 ) ! "r & left indices co9cid+1 :5 opposite any angle on ,t w be f\nd its tang5t on ,a4 ,f :at has be5 said above1 x foll[s t ! tang5ts read on ! ,l4,h4 scale ( ,a h values ext5d+ f #jja to #ja2 :ile ^? read on ! ,r4,h4 scale ( ,a h values f #ja to #aj4 ,o!rwise express$1 to ! values ( any tang5t read on ! ,l4,h4 scale ( ,a a cyph] is to be prefix$2 :ile if f\nd on ! ,r4,h4 scale1 x is read directly z a decimal4 ,,ex4--,f9d tan4 #c #ej4 ,plac+ ! slide z direct$1 ! read+ on ,a opposite #c #ej on ,t is f\nd to be #fg4 ,z ?is is f\nd on ! ,l4,h4 scale ( ,a1 x is to be read z #jjfg4 ,,ex4--,f9d tan4 #ag #de4 ,"h ! read+ on ,a opposite #ag #de on ,t is #cb1 & z x is f\nd on ! ,r4,h4 scale ( ,a x is read z #jcb4 ,z 9 ! case ( ! scale ( s9es1 ! tang5ts may be f\nd )\t rev]s+ ! slide1 :5 a fix$ index is provid$ 9 ! back ( ! rule = ! ,t scale4 ,we rev]t n[ to a consid]ation ( ^? rules 9 :i* a s+le tang5t scale is provid$4 ,x w be und]/ood t 9 ?is case ! slide is plac$ s t ! scale ,t is adjac5t to ! ,d scale1 & t :5 ! indices ( bo? are plac$ 9 agreem5t1 ! value ( ! tang5t ( any angle on ,t 7f #e #dc to #de7 may be read (f on ,d1 ! result s f\nd be+ read z :olly decimal4 ,?us tan4 #ac #bj is read #jbcg4 ,if a back index is provid$1 ! slide is us$ 9 its normal position1 :51 sett+ ! angle on ! tang5t scale to ?is index1 ! result c be read on ,c ov] ! ,l4,h4 index ( ,d4 ,! tang5ts ( angles above #de are obta9$ by ! =mula3 ,tan4 7#a7_/7tan4 7#ij 774 ,= all angles f #de to 7#ij #e #dc7 we proce$ z foll[s3--,place 7#ij 7 on ,t to ! ,r4,h4 index ( ,d1 & read tan4 on ,d "u ! ,l4,h4 index ( ,t4 ,! fir/ figure 9 ! value ?us obta9$ is to be read z an integ]4 ,?us1 to f9d tan4 #ga #bj we place #ij #ga #bj #ah #dj on ,t1 to ! ,r4,h4 index ( ,d1 & "u ! ,l4,h4 index ( ,t read #bif1 ! requir$ tang5t4 ,! tang5ts ( angles less ?an #dj are s5sibly proportional to ! angles !mselves1 & z !y may !re=e be consid]$ z s9es1 _! value is det]m9$ by ! aid ( ! s+le & d\ble acc5t m>ks on ! s9e scale1 z previ\sly expla9$4 ,! rules = ! numb] ( integ]s are ! same z = ! s9es4 ,multiplication & division ( tang5ts may be q readily effect$4 ,,ex4--,tan4 #ba #ej #ae #f4 ,set ,l4,h4 index ( ,t to #ae on ,d1 & "u #ba #ej on ,t read #f on ,d4 ,,ex4--,tan4 #gb #dj #aag #cge4 ,set 7#ij #gb #dj7 #ag #bj on ,t to #aag on ,d1 & "u ,r4,h4 index ( ,t read #cge on ,d4 ,cos9es ( ,angles4--,! cos9es ( angles may be det]m9$ by plac+ ! scale ,s ) its indices co9cid+ ) ^? ( ,a1 & :5 opposite 7#ij 7 on ,s is read cos4 on ,a4 ,if ! result is read on ! ,l4,h4 scale ( ,a1 a cyph] is to be prefix$ to ! value read2 :ile if x is read on ! ,r4,h4 scale ( ,a1 ! value is read directly z a decimal4 ,?us1 to det]m9e cos4 #hf #cj we f9d opposite 7#ij #hf #cj7 #c #cj on ,s1 #fa on ,a1 & z ?is is on ! ,l4,h4 scale ! result is read #jjfa4 ,aga91 to f9d cos4 #ei #bj we read opposite 7#ij #ei #bj7 or #cj #dj on ,s1 #ea on ,a1 & z ?is is f\nd on ! ,r4,h4 scale ( ,a1 x is read #jea4 ,9 f9d+ ! cos9es ( small angles x w be se5 t direct read+ on ! rule becomes impossible = angles ( less ?an #bj4 ,x is advisable 9 su* cases to adopt ! me?od describ$ = det]m9+ ! s9es ( ! l>ge angles ( :i* ! complem5ts are s\ght4 ,cotang5ts ( ,angles4--,f ! me?ods ( f9d+ ! tang5ts ( angles previ\sly describ$1 x w be app>5t t ! cotang5ts ( angles may also be obta9$ ) equal facility4 ,= angles betwe5 #e #de & #de1 ! proc$ure is ! same z t = f9d+ tang5ts ( angles great] ?an #de4 ,?us1 ! angle on scale ,t is br\ght to ! ,r4,h4 index ( ,d1 & ! cotang5t read (f on ,d "u ! ,l4,h4 index ( ,t4 ,! fir/ figure ( ! result s f\nd is to be read z an integ]4 ,if ! angle 77 lies betwe5 #de & #hd #ae1 ! slide is plac$ s t ! indices ( ,t co9cide ) ^? ( ,d1 & ! result is !n read (f on ,d opposite 7#ij 7 on ,t4 ,9 ?is case ! value is :olly decimal4 ,secants ( ,angles4--,! secants ( angles are readily f\nd by br++ 7#ij 7 on ,s to ! ,r4,h4 index ( ,a & read+ ! result on ,a ov] ! ,l4,h4 index ( ,s4 ,if ! value is f\nd on ! ,l4,h4 scale ( ,a1 ! fir/ figure is to be read z an integ]2 :ile if ! result is read on ! ,r4,h4 scale ( ,a1 ! fir/ two figures are to be reg>d$ z integ]s4 ,cosecants ( ,angles4--,! cosecants ( angles are f\nd by plac+ ! angle on ,s to ! ,r4,h4 index ( ,a1 & read+ ! value f\nd on ,a ov] ! ,l4,h4 index ( ,s4 ,if ! result is read on ! ,l4,h4 scale ( ,a1 ! fir/ figure is to be read z an integ]2 :ile if ! result is f\nd on ! ,r4,h4 scale ( ,a1 ! fir/ two figures are to be read z integ]s4 ,x w be not$ t "s ( ! rules "h giv5 = det]m9+ ! sev]al trigonometrical functions ( angles apply only to ^? =ms ( rules 9 :i* a s+le scale ( tang5ts ,t is us$1 read+ f left to "r4 ,= ! o!r arrangem5ts ( ! scale1 previ\sly ref]r$ to1 "s slight modification ( ! me?od ( proc$ure 9 f9d+ ! tang5ts & cotang5ts ( angles w be necess>y2 b z 9 ea* case ! nature & ext5t ( ?is modification is evid5t1 no fur!r directions are requir$4 ,,! ,,solution ,,( ,,"r-,,angl$ ,,triangles4 ,f ! =ego+ explanation ( ! mann] ( det]m9+ ! trigonometrical functions ( angles1 ! me?ods ( solv+ "r-angl$ triangles w be readily p]ceiv$1 & only a few examples ne$ !re=e be giv54 ,let a & b repres5t ! sides & c ! hypo!nuse ( a "r-angl$ triangle1 & a & b ! angles opposite to ! sides4 ,!n ( ! possible cases we w take 7#a47 ,giv5 c & a1 to f9d a1 b1 & b4 ,! angle b #ij a1 :ile a c s9 a & b c s9 b4 ,to f9d a1 !re=e1 ! index ( ,s is set to c on ,a1 & ! value ( a read on ,a opposite a on ,s4 ,9 ! same mann] ! value ( b is obta9$4 ,,ex4--,giv5 9 a "r-angl$ triangle c #i ft4 & a #cj4 ,f9d a1 b1 & b4 ,! angle b #ij #cj #fj4 ,to f9d a1 set ,r4,h4 index ( ,s to #i on ,a1 & ov] #cj on ,s read a #de ft4 on ,a4 ,also1 ) ! slide 9 ! same position1 read b #gh ft4 #ggid on ,a ov] #fj on ,s4 7#b47 ,giv5 a & c1 to det]m9e a1 b1 & b4 ,9 ?is case advantage is tak5 ( ! fact t 9 e triangle ! sides are proportional to ! s9es ( ! opposite angles4 ,!re=e1 z 9 ?is case ! hypo!nuse c subt5ds a "r angle1 ( :i* ! s9e #a1 ! ,r4,h4 index 7or #ij7 on ,s is set to ! l5g? ( c on ,a1 :5 "u a on ,a is f\nd a on ,s4 ,h5ce b & b may be det]m9$4 7#c47 ,giv5 a & a1 to f9d b1 c1 & b4 ,"h b 7#ij a71 & ! solution is simil> to ! =ego+4 7#d47 ,giv5 a & b1 to f9d a1 b1 & c4 ,to f9d a1 we h tan4 a a_/b1 :i* 9 ! above example w be 7#de7_/7gh7 #jegg4 ,!re=e1 plac+ ! slide s t ! indices ( ,t co9cide ) ^? ( ,d1 we read opposite #jegg on ,d ! value ( a #cj4 ,! hypo!nuse c is readily obta9$ f c a_/7s9 a74 ,,! ,,solution ,,( ,,oblique-,,angl$ ,,triangles4 ,us+ ! same lett]s z be=e to designate ! ?ree sides & ! subt5d+ angles ( oblique-angl$ triangles1 we h ! foll[+ cases3-- 7#a47 ,giv5 "o side & two angles1 z a1 a1 & b1 to f9d b1 c1 & c4 ,9 ! fir/ place1 c #ahj 7a b72 also we note t1 z ! sides are proportional to ! s9es ( ! opposite angles1 b 7a s9e b7_/7s9e a7 & c 7a s9e c7_/7s9e a74 ,tak+ z an example1 a #de1 a #eg1 & b #fc1 we h c #ahj 7#eg #fc7 #fj4 ,to f9d b & c1 set a on ,s to a on ,a1 & read (f on ,a above #fc & #fj ! values ( b 7 #dgh7 & c 7 #dfd7 respectively4 7#b47 ,giv5 a1 b1 & a1 to f9d b1 c1 & c4 ,9 ?is case ! angle a on ,s is plac$ "u ! l5g? ( side a on ,a & "u b on ,a is f\nd ! angle b on ,s4 ,! angle c #ahj 7a b71 :5ce ! l5g? c c be read (f on ,a ov] c on ,s4 7#c47 ,giv5 ! sides & ! includ$ angle1 to f9d ! o!r side & ! rema9+ angles4 ,if1 = example1 !re are giv5 a #fe1 b #db1 & ! includ$ angle c #ee1 we h 7a b7 7a b7 tan4 7a b7_/7#b7 tan4 7a b7_/7#b74 ,!n1 s9ce a b #ahj #ee #abe1 x foll[s t 7a b7_/7#b7 7#abe7_/7b7 #fb #cj4 ,by ! rule = tang5ts ( angles great] ?an #de1 we f9d tan4 #fb #cj #aib4 ,ins]t+ 9 ! above proportion ! values ?us f\nd1 we h #ajg #bc #aib tan4 7a b7_/7#b74 ,f ?is x is f\nd t ! value ( ! tang5t is #jdab1 & plac+ ! slide ) all indices co9cid+1 x is se5 t ?is value on ,d corresponds to an angle ( #bb #be4 ,!re=e1 s9ce 7a b7_/7#b7 #fb #cj1 & 7a b7_/7#b7 #bb #be1 x foll[s t a #hd #ee1 & b #dj #e4 ,f9ally1 to det]m9e ! side c1 we h c 7a s9 c7_/7s9 a7 z be=e4 ,,practical ,,trigonometrical ,,applications4 ,a few examples illu/rative ( ! application ( ! me?ods ( det]m9+ ! functions ( angles1 etc41 describ$ 9 ! prec$+ section1 w n[ be giv54 ,to f9d ! *ord ( an arc1 hav+ giv5 ! includ$ angle & ! radius4 ,) ! slide plac$ 9 ! rule ) ! ,c & ,d scales outw>d1 br+ "o-half ( ! giv5 angle on ,s to ! index m>k 9 ! back ( ! rule1 & read ! *ord on ,b "u twice ! radius on ,a4 ,,ex4--,requir$ ! *ord ( an arc ( #ae1 ! radius be+ #bc 94 ,set #g #cj on ,s to ! index m>k 9 ! back ( ! rule1 & "u #df on ,a read #f 941 ! requir$ l5g? ( *ord on ,b4 ,to f9d ! area ( a triangle1 giv5 two sides & ! includ$ angle4 ,set ! angle on ,s to ! index m>k on ! back ( ! rule1 & br+ cursor to #b on ,b4 ,!n br+ ! l5g? ( "o side on ,b to cursor1 cursor to #a on ,b1 ! l5g? ( ! o!r side on ,b to cursor1 & read area on ,b "u index ( ,a4 ,,ex4--,! sides ( a triangle are #e & #f ft4 9 l5g? respectively1 & !y include an angle ( #bj4 ,f9d ! area4 ,set #bj on ,s to index m>k1 br+ cursor to #b on ,b1 #e on ,b to cursor1 cursor to #a on ,b1 #f on ,b to cursor1 & "u #a on ,a read ! area #eac sq4 ft4 on ,b4 ,to f9d ! numb] ( degrees 9 a gradi5t1 giv5 ! rise p] c5t4 ,place ! slide ) ! indices ( ,t co9cid5t ) ^? ( ,d1 & ov] ! rate p] c5t4 on ,d read numb] ( degrees 9 ! slope on ,t4 ,z ! arrangem5t ( rule we h *iefly consid]$ has only a s+le ,t scale1 x w be se5 t only solutions ( ! above problem involv+ slopes betwe5 #aj & #ajj p] c5t4 c be directly read (f4 ,= small] angles1 "o ( ! =mul = ! det]m9ation ( ! tang5ts ( submultiple angles mu/ be us$4 ,9 rules hav+ a d\ble ,t scale 7:i* is us$ ) ! ,a scale7 ! value 9 degrees ( any slope f #a to #ajj p] c5t4 c be directly read (f on ,a4 ,to f9d ! numb] ( degrees1 :5 ! gradi5t is express$ z #a 9 x4 ,place ! index ( ,t to x on ,d1 & ov] index ( ,d read ! requir$ angle 9 degrees on ,t4 ,,ex4--,f9d ! numb] ( degrees 9 a gradi5t ( #a 9 #ch4 ,set #a on ,t to #ch on ,d1 & ov] ,r4,h4 index ( ,d read #ad #de on ,t4 ,giv5 ! lap1 ! lead & ! travel ( an eng9e slide valve1 to f9d ! angle ( advance4 ,set 7lap lead7 on ,b to half ! travel ( ! valve on ,a1 & read ! angle ( advance on ,s at ! index m>k on ! back ( ! rule4 ,,ex4--,valve travel #d;941 lap #a 941 lead 94 ,f9d angle ( advance4 ,set #a #acab on ,b to #bbe on ,a1 & read #ce #dj on ,s opposite ! index on ! back ( ! rule4 ,giv5 ! angul> advance 1 ! lap & ! travel ( a slide valve1 to f9d ! cut-(f 9 p]c5tage ( ! /roke4 ,place ! lap on ,b to half ! travel ( valve on ,a1 & read on ,s ! angle 7! supplem5t ( ! angle ( ! ecc5tric7 f\nd opposite ! index 9 ! back ( ! rule4 ,to ?is angle1 add ! angle ( advance & d$uct ! sum f #ahj1 ?us obta9+ ! angle ( ! crank at ! po9t ( cut-(f4 ,to ! cos9e ( ! supplem5t ( ?is angle1 add #a & multiply ! result by #ej1 obta9+ ! p]c5tage ( /roke complet$ :5 cut-(f occurs4 ,,ex4--,giv5 ! angul> advance #ce #dj1 ! valve travel #d;941 & ! lap #a 941 f9d ! angle ( ! crank at cut-(f & ! admission p]iod express$ z a p]c5tage ( ! /roke4 ,set #a on ,b to #bbe on ,a1 & read (f on ,s opposite ! index1 ! supplem5t ( ! angle ( ! ecc5tric #bf #bj4 ,!n #ahj 7#ce #dj #bf #bj7 #aah ! crank angle at ! po9t ( cut-(f4 ,fur!r1 cos4 #aah cos4 #fb s9 7#ij #fb7 s9 #bh1 & plac+ #bh on ,s to ! back index1 ! cos9e1 read on ,b "u ,r4,h4 index ( ,a1 is f\nd to be #jdfi4 ,add+ #a & plac+ ! ,l4,h4 index ( ,c to ! result1 #adfi1 on ,d1 we read (f "u #ej on ,c1 ! requir$ p]iod ( admission #gcd p] c5t4 on ,d4 ,! trigonometrical scales are useful = evaluat+ c]ta9 =mul4 ,?us 9 ! foll[+ expressions1 if we f9d ! angle a su* t s94 a k1 we c write3-- 7k7_/7#a k#b7 tan4 a2 7#a k#b7_/7k7 cot4 a2 7#a k#b7 cos4 a2 etc4 ,9 ! fir/ expression1 take k #jbih4 ,place ! slide ) ! s9e scale outw>d & ) its indices agree+ ) ! indices ( ! rule4 ,set ! cursor to #jbih on ! 7,r4,h47 scale ( ,a1 & read #ag #bj on ! s9e scale z ! angle requir$4 ,!n "u #ag #bj on ! tang5t scale1 read #jcab on ,d z ! result4 ,,slide ,,rules ,,) ,,log4-,,log4 ,,scales4 ,= occasional requirem5ts1 ! me?od describ$ on page #de ( det]m9+ p[]s & roots o!r ?an ! squ>e & cube1 is q satisfactory4 ,:51 h[ev]1 a numb] ( su* calculations are to be made1 ! process may be simplifi$ consid]ably by ! use ( :at are kn[n z log4-log41 logo-log41 or logometric scales1 9 conjunction ) ! ord9>y scales ( ! rule4 ,! pr9ciple involv$ w be und]/ood f a consid]ation ( ^? rules = log>i?mic computation 7page #h7 :i* ref] to p[]s & roots4 ,f ^! x is se5 t :ile = ! multiplication & division ( numb]s we add _! log>i?ms1 = involution & evolution we require to multiply or divide ! log>i?ms ( ! numb]s by ! expon5t ( ! p[] or root z ! case may be4 ,?us to f9d #cb4c1 we h 7log4 #c7 #bc log4 x1 & by ! ord9>y me?od describ$ on page #de we %\ld det]m9e log4 #c by ! aid ( ! scale ,l on ! back ( ! slide1 multiply ?is by #bc by us+ ! ,c & ,d scales 9 ! usual mann]1 transf] ! result to scale ,l1 & read ! value ( x on ,d "u #a on ,c4 ,by ! simpl] me?od1 fir/ propos$ by ,dr4 ,p4 ,m4 ,roget1#h ! multiplication ( log4 #c by #bc is effect$ 9 ! same way z ) any two ord9>y factors--i4e41 by add+ _! log>i?ms & f9d+ ! numb] correspond+ to ! result+ log>i?m4 ,9 ?is case we h log4 7log4 #c7 log4 #bc log4 7log4 x74 ,! fir/ ( ! ?ree t]ms is obvi\sly ! log>i?m ( ! log>i?m ( #c1 ! second is ! simple log>i?m ( #bc1 & ! ?ird ! log>i?m ( ! log>i?m ( ! answ]4 ,h5ce1 if we h a scale s graduat$ t ! di/ances f ! po9t ( orig9 repres5t ! log>i?ms ( ! log>i?ms 7! log4-logs47 ( ! numb]s engrav$ ^u x1 !n by us+ ?is 9 conjunction ) ! ord9>y scale ( log>i?ms1 we c effect ! requir$ multiplication 9 a mann] :i* is bo? exp$iti\s & conv5i5t4 ,slightly v>y+ arrangem5ts ( ! log4-log4 scale1 sometimes ref]r$ to z ! 8,p l9e10 h be5 introduc$ f "t to "t1 b latt]ly ! increas+ use ( expon5tial =mul 9 !rmodynamic1 electrical1 & physical calculations has l$ to a revival ( int]e/ 9 ,dr4 ,roget's inv5tion1 & v>i\s arrangem5ts ( rules ) log4-log4 scales are n[ available4 ,! ,davis ,log4-,log4 ,rule4--,9 ! rule introduc$ by ,messrs4 ,john ,davis ,son ,limit$1 ,d]by1 ! log4-log4 scales are plac$ ^u a sep>ate slide--a plan :i* has ! advantage ( leav+ ! rule intact = all ord9>y purposes1 :ile provid+ a l5g? ( #dj 94 = ! log4-log4 scales4 ,9 ! #aj 94 ,davis rule "o face ( ! slide1 m>k$ ,e1 has two log4-log4 scales = numb]s great] ?an unity1 ! l[] ext5d+ f #ajg to #b1 & ! upp] cont9u+ ! graduations f #b to #ajjj4 ,on ! rev]se face ( ! slide1 m>k$ -,e1 are two log4-log4 scales = numb]s less ?an unity1 ! upp] ext5d+ f #jjja to #je1 & ! l[] cont9u+ ! graduations f #je to #jicc4 ,bo? sets ( scales are us$ 9 conjunction ) ! l[] or ,d scale ( ! rule1 :i* is to be prim>ily reg>d$ z runn+ f #a to #aj1 & con/itutes a scale ( expon5ts4 ,9 ! #bj 94 rule ! log4-log4 scales are m ext5sive1 & are us$ 9 conjunction ) ! upp] or ,a scale ( ! rule 7#a to #ajj72 9 :at foll[s1 h[ev]1 ! #aj 94 rule is m p>ticul>ly ref]r$ to4 ,x has be5 expla9$ t on ! log4-log4 scale ! di/ance ( any numb]$ graduation f ! po9t ( orig9 repres5ts ! log4-log4 ( ! numb]4 ,! po9t ( orig9 w obvi\sly be t graduation ^: log4-log4 #j4 ,?is is se5 to be #aj1 s9ce log4 7log4 #aj7 log4 #a #j4 ,h5ce1 conf9+ att5tion to ! ,e scale1 to locate ! graduation #bj1 we h log4 7log4 #bj7 log4 #acja #jaacig1 s t if ! scale ,d is #be cm4 long1 ! di/ance betwe5 #aj & #bj on ! correspond+ log4-log4 scale w\ld be #aacig #d #bhdi mm4 ,= numb]s less ?an #aj ! result+ log4-logs4 w be negative1 & ! di/ances w be spac$ (f f ! po9t ( orig9 9 a negative direction--i4e41 f "r to left4 ,?us1 to locate ! graduation #e1 we h log4 7log4 #e7 log4 #jfii #ahdd2 i4e41 #a #jhdd or #jaef2 s t ! graduation m>k$ #e w\ld be plac$ #aef #d #ci mm4 di/ant f #aj 9 a negative direction1 & proce$+ 9 a simil> mann]1 ! scale may be ext5d$ 9 ei!r direction4 ,9 ! -,e scale1 ! notation runs 9 ! rev]se direction to t ( ! ,e scale1 b 9 all o!r respects x is precisely analog\s1 ! di/ance f ! po9t ( orig9 7#ja 9 ?is case7 to any graduation x repres5t+ log4 -log4 x44 ,x foll[s t ( ! simil>ly situat$ graduations on ! two scales1 ^? on ! -,e scale are ! reciprocals ( ^? on ! ,e scale4 ,?is may be readily v]ifi$ by sett+1 say1 #aj on ,e to 7,r4,h47 #a on ,d1 :5 turn+ to ! back ( ! rule we f9d #ja on -,e agree+ ) ! index m>k 9 ! ap]ture at ! "r-h& extremity ( ! rule4 ,9 us+ ! log4-log4 scales x is important to obs]ve 7#a7 t ! values engrav$ on ! scale are def9ite & unalt]able 7e4g41 #ab c only be read z #ab & n z #abj1 #jjjab1 etc41 z ) ! ord9>y scales72 7#b7 t ! upp] portion ( ea* scale %\ld be reg>d$ z =m+ a prolongation to ! "r ( ! l[] portion2 & 7#c7 t imm$iately above any value on ! l[] portion ( ! scale is f\nd ! #aj? p[] ( t value on ! upp] portion ( ! scale4 ,keep+ ^! po9ts 9 view1 if we set #aa on ,e to #a on ,d we f9d ov] #b on ,d ! value ( #aab #aba on ,e4 ,simil>ly1 ov] #c we f9d #aac #acca1 & s on4 ,!n1 read+ across ! slide1 we h1 ov] #b1 ! value ( #aab #aj #aabj #fgc1 & ov] #c we h #aac #aj #aacj #agd4 ,h5ce ! rule3--,to f9d ! value ( xn1 set x on ,e to #a on ,d1 & ov] n on ,d read xn on ,e4 ,) ! slide set z above1 ! #h?1 #i?1 etc41 p[]s ( #aa _c be read (f2 b x is se5 t1 accord+ to 7#b7 9 ! =ego+1 ! miss+ portion ( ! ,e scale is t "p ( ! upp] scale 7#b to ab\t #bf7 :i* is outside ! rule to ! left4 ,h5ce plac+ #aa to #aj on ,d1 ! #h?1 #i?1 etc41 p[]s ( #aa w be read (f on ! upp] "p ( ! ,e scale4 ,9 g5]al1 !n1 ,if x on ! l[] l9e is set to #a on ,d1 !n xn is read directly on t l9e & x#ajn on ! upp] l9e4 ,if x on ! upp] l9e is set to #a on ,d1 !n xn is read directly on t l9e & x on ! l[] l9e4 ,if x on ! l[] l9e is set to #aj on ,d1 !n x is read directly on t l9e & xn on ! upp] l9e4 ,if x on ! upp] l9e is set to #aj on ,d1 !n x is read directly on t l9e & x on ! l[] l9e4 ,^! rules are conv5i5tly exhibit$ 9 ! accompany+ diagram 7,fig4 #ad74 ,!y are equally applicable to bo? ! ,e & -,e scales ( ! #aj 94 rule1 & include practically all ! in/ruction requir$ = det]m9+ ! n? p[] or ! n? root ( a numb]4 ,!y d n apply directly to ! #bj 94 rule1 h[ev]1 = "h ! relation ( ! l[] & upp] scales w be xn & x#ajjn4 ,,ex4--,f9d #aafgbef4 ,set #aafg on ,e to #a on ,d1 & ov] #bef on ,d read #adhe on ,e4 ,,ex4--,f9d #dfafa4 ,set #df on upp] ,e scale to #a on ,d1 & ov] #afa on ,d read #aag 7#aafg7 on ,e4 ,,ex4--,f9d #adjbg & #adbg4 ,set #ad on ,e to #aj on ,d1 & ov] #bg on ,d read #ajie #adjbg on l[] ,e scale & #bdh #adbg on upp] ,e scale4 ,illu/ration3 ,,fig4 #ad4 ,,ex4--,f9d #dfjjahd & #dfjahd4 ,set #df on upp] ,e scale to #aj on ,d1 & ov] #ahd on ,d read #ajgc on l[] ,e scale & #bjbb 7#bjbbh7 on upp] ,e scale4 ,,ex4--,f9d #jjgdaae4 ,us+ ! -,e scale1 set #jjgd to #a on ,d1 & ov] #aae on ,d read #jje on -,e4 ,! me?od ( det]m9+ ! root ( a numb] w be obvi\s f ! prec$+ examples4 ,,ex4--,f9d #a4d7ag7 & #ad7ag74 ,set #ag on ,e to #ad on ,d1 & ov] #a on ,d read #gef on upp] ,e scale & #abbd on l[] ,e scale4 ,,ex4--,f9d #jjca7jiad74 ,set #jiad on -,e to #ca on ,d1 & ov] #aj on ,d read #jjee on upp] -,e scale4 ,:5 ! expon5t n is fractional1 x is (t5 possible to obta9 ! result directly ) "o sett+ ( ! slide4 ,?us to det]m9e #aace by ! fir/ me?od we f9d #ajfbe1 & plac+ #aace on ,e to #a on ,d1 read #aadd on ,e ov] #ajfbe on ,d4 ,by ! direct me?od we place #aace on ! ,e scale on #af on ,d1 & ov] #ag on ,d read #aadd on ,e4 ,x w be se5 t s9ce ! scale ,d is assum$ to run f #a to #aj we are unable to read #af & #ag on ?is scale2 b x is obvi\s t ! ratios 7#ag7_/7af7 & 7#ag7_/7af7 are id5tical1 & x is ) ! ratio only t we are1 9 effect1 conc]n$4 ,s9ce an expression ( ! =m x-n 7#a7_/7xn7 or 77#a7_/7x77n1 ! requir$ value may be obta9$ by fir/ det]m9+ ! reciprocal ( x & proce$+ z be=e4 ,by us+ bo? ! direct & reciprocal log4-log4 scales 7,e & -,e7 9 conjunction h[ev]1 ! requir$ value c be read directly f ! rule1 & ! prelim9>y calculation entirely avoid$4 ,9 ! ,davis =m ( rule1 ! result c be read on ! -,e scale1 us$ 9 conjunction ) ! ,d scale ( ! rule1 x on ,e be+ set to ! index m>k 9 ! ap]ture 9 ! back ( ! rule4 ,,ex4--,f9d ! value ( #aaieafe4 ,set #aaie on ,e to ! index 9 ! left ap]ture 9 ! back ( ! rule1 & ov] #afe on ,d read #jgde on ! -,e scale4 ,x may be not$ 9 pass+ t ! log4-log4 scale af=ds a simple means = det]m9+ ! log>i?m or anti-log>i?m ( a numb] to any base4 ,= ?is purpose x is necess>y to set ! base ( ! giv5 sy/em on ,e to #a on ,d1 :5 "u any numb] on ,e w be f\nd its log>i?m on ,d4 ,?us1 = common logs41 we set ! base #aj on ,e to #a on ,d1 & "u #ajj we f9d #b1 ! requir$ log4 ,simil>ly we read log4 #bj #acja2 log4 #ee #agd2 log4 #eej #bgd1 etc4 ,read+ rev]sely1 ov] #ach on ,d we f9d its antilog4 #bd on ,e2 also antilog4 #aeh #ch2 antilog4 #aai #aee1 etc4 ,= logs4 ( numb]s "u #aj we set ! base #aj to #aj on ,d2 h5ce ! read+s on ,d w be read z "o-t5? _! app>5t value4 ,?us log4 #c #jdgg2 log4 #ebe #jgb2 antilog4 #jdae #bf2 antilog4 #jebe #c4ce1 etc4 ,! logs4 ( ! numb]s on ! l[] half ( ! ,e scale w also be f\nd on ! ,d scale2 b a consid]ation ( ,fig4 #ad w %[ t ?is w be read z "o-t5? its face value if ! base is set to #a on ,d1 & z "o-hundr$? if ! base is set to #aj4 ,= natural1 hyp]bolic1 or ,napi]ian log>i?ms1 ! base is #bgah4 ,a special l9e m>k$ or e s]ves to locate ! exact position ( ?is value on ! ,e scale1 & plac+ ?is to #a on ,d we read log4e #dce #adg2 log4e #gd #bj2 antilog4e #bhi #ah1 etc4 ,! o!r p>ts ( ! scale are read z already describ$ = common logs4 ,calculations involv+ p[]s ( e are frequ5tly met )1 & ^! are facilitat$ by us+ ! special graduation l9e ref]r$ to1 z w be readily und]/ood4 ,if x is requir$ to det]m9e ! p[] or root ( a numb] :i* does n appe> on ei!r ( ! log4-log4 scales1 we may break up ! numb] into factors4 ,usually x is conv5i5t to make "o ( ! factors a p[] ( #aj4 ,,ex4--#ciejaig #cieaig #ajc #aig #cieaig #ajeia4 ,!n #cieaig #ae1 & #ajeia 7or antilog47 #eia #hab1jjj4 ,h5ce1 #ae #hab1jjj #ab1ahj1jjj is ! result s\ght4 ,numb]s :i* are to be f\nd 9 ! high] "p ( ! log4-log4 scale may (t5 be factoris$ 9 ?is way1 & great] accuracy obta9$ ?an by direct read+4 ,! =m ( log4-log4 rule :i* has be5 ma9ly dealt ) 9 ! =ego+ gives a scale ( comp>atively long range1 & ! only objection to ! arrangem5t adopt$ is ! use ( a sep>ate slide4 ,! ,jackson-,davis ,d\ble ,slide ,rule4--,9 ?is in/rum5t a pair ( alum9ium clips enable ! log4-log4 slide to be tempor>ily atta*$ to ! l[] edge ( ! ord9>y rule1 & us$1 by means ( a special cursor1 9 conjunction ) ! ,c scale ( ! ord9>y slide4 ,9 ?is way bo? ! log4-log4 & ord9>y scales are available )\t ! tr\ble ( replac+ "o slide by ! o!r4 ,s9ce ! scale ( expon5ts is n[ on ! slide1 ! value ( xn w be obta9$ by sett+ #a on ,c to x on ,e & read+ ! result on ,e "u n on ,c4 ,by us+ a pair ( log4-log4 slides1 "o 9 ! rule & "o clamp$ to ! edge by ! clips1 we h an arrangem5t :i* is v useful 9 d$uc+ empirical =mul ( ! type y xn4 ,! ,yokota ,slide ,rule4--,9 ?is in/rum5t ! log4-log4 scales are plac$ on ! face ( ! rule1 ea* set compris+ ?ree l9es4 ,^!1 = numb]s great] ?an #a1 are f\nd above ! ,a scale :ile ! ?ree reciprocal log4-log4 l9es are bel[ ! ,d scale4 ,bo? sets are us$ 9 conjunction ) ! ,c scale on ! slide4 ,o!r features ( ?is rule are3--,! ord9>y scales are #aj 94 long in/ead ( #be cm4 z hi!rto usual2 h5ce ! log>i?ms ( numb]s c be read on ! ord9>y scale ( in*es on ! edge ( ! rule4 ,!re is a scale ( cubes 9 ! c5tre ( ! slide & on ! back ( ! slide !re is a scale ( secants 9 addition to ! s9e & tang5t scales4 ,illu/ration3 ,,fig4 #ae4 ,! ,fab] ,log4-log4 ,rule4--,9 ?is in/rum5t %[n 9 ,fig4 #ae1 ! two log4-log4 scales are plac$ on ! face ( ! rule4 ,"o section1 ext5d+ f #aa to #bi1 is plac$ above ! ,a scale1 & ! o!r section1 ext5d+ f #bi to #ajj1jjj1 is plac$ bel[ ! ,d scale4 ,^! scales are us$ 9 conjunction ) ! ,c scale ( ! slide 9 ! mann] previ\sly describ$4 ,! wid? ( ! rule is increas$ slightly1 b ! arrangem5t is m conv5i5t ?an t =m]ly employ$1 :]e9 ! log4-log4 scales 7 plac$ on ! bevell$ edge ( ! rule & read by a tongue project+ f ! cursor4 ,illu/ration3 ,,fig4 #af4 ,ano!r novel feature ( ?is rule is ! provision ( two special scales at ! bottom ( ! groove1 to :i* a bevell$ metal index or m>k] on ! left end ( ! slide c be set4 ,! upp] ( ^! scales is = det]m9+ ! effici5cy ( dynamos & electric motors2 ! l[] = det]m9+ ! loss ( pot5tial 9 an electric circuit4 ,! ,p]ry ,log4-log4 ,rule4--,9 ?is rule1 introduc$ by ,messrs4 ,a4 ,g4 ,?ornton1 ,limit$1 ,man*e/]1 ! log4-log4 scales are arrang$ z 9 ,fig4 #af1 ! ,e scale1 runn+ f #aa to #aj1jjj1 be+ plac$ above ! ,a scale ( ! rule1 & ! -,e or ,e#a scale runn+ f #jic to #jjjja1 bel[ ! ,d scale ( ! rule4 ,^! scales are read 9 conjunction ) ! ,b scales on ! slide by ! aid ( ! cursor4 ,! foll[+ tabul> /atem5t embodies all ! in/ructions requir$ = us+ ?is =m ( log4-log4 slide rule3-- ,:5 x is great] ?an #a4 xn ,set #a on ,b to x on ,e2 ov] n on ,b read xn on ,e x-n ,set #a on ,b to x on ,e2 "u n on ,b read x-n on ,e#a x ,set n on ,b to x on ,e2 ov] #a on ,b read x on ,e x ,set n on ,b to x on ,e2 "u #a on ,b read x on ,e#a ,:5 x is less ?an #a4 xn ,set #a on ,b to x on ,e#a2 "u n on ,b read xn on ,e#a x-n ,set #a on ,b to x on ,e#a2 ov] n on ,b read x-n on ,e x ,set n on ,b to x on ,e#a2 "u #a on ,b read x on ,e#a x ,set n on ,b to x on ,e#a2 ov] #a on ,b read x on ,e ,if #aj on ,b is us$ 9 place ( #a on ,b1 read x 9 place ( xn on ,e1 & x- 9 place ( x-n on ,e#a4 ,if #ajj on ,b is us$1 ^! read+s are to be tak5 z x & x- respectively4 ,9 rules ) no -,e scale ! value ( x-n is obta9$ by ! usual rules = reciprocals4 ,we may ei!r det]m9e xn & f9d its reciprocal or1 fir/ f9d ! reciprocal ( x & raise x to ! n? p[]4 ,! fir/ me?od %\ld be foll[$ :5 ! numb] x is f\nd on ! ,e scale4 ,,ex4--#cdeahb #jaje4 ,set #a on ,c to #cde on ,e1 & "u #ahb on ,c read #iea on ,c4 ,!n set #a on ,b to #ie on ,a1 & "u index ( ,a read #jaje on ,b4 ,:5 x is less ?an #a ! second me?od is m suitable4 ,,ex4--#jbcagg 77#a7_/7jbc77agg #dceagg #ace ,set #a on ,b to #jbc on ,a1 & "u index ( ,a read 7#a7_/7jbc7 #dce on ,b4 ,set #a on ,c to #dce on ,e1 & "u #agg on ,c read #ace on ,e4 ,z ) ! ,davis rule1 ! expon5t scale ,c w be read z ? its face value if its ,r4,h4 index 7#aj7 is us$ 9 place ( #a4 ,,special ,,types ,,( ,,slide ,,rules4 ,9 addition1 to ! new =ms ( log4-log4 slide rules previ\sly describ$1 sev]al o!r arrangem5ts h be5 rec5tly introduc$1 notably a s]ies by ,mr4 ,a4 ,ne/l]1 ( ,lahr 7,london3 ,a4 ,fa/l+]1 ,sn[ ,hill74 ,^! comprise ! 8,rietz10 ! 8,precision10 ! 8,univ]sal10 & ! 8,fix0 slide rules4 ,,! ,,rietz ,,rule4--,9 ?is rule ! usual scales ,a1 ,b1 ,c1 & ,d1 are provid$1 :ile at ! upp] edge is a scale1 :i*1 be+ ?ree times ! range ( ! ,d scale1 enables cubes & cube roots to be directly evaluat$ & also n & n4 ,a scale at ! l[] edge ( ! rule gives ! mantissa ( ! log>i?ms ( ! numb]s on ,d4 ,,! ,,precision ,,slide ,,rule4--,9 ?is rule ! scales are s arrang$ t ! accuracy ( a #bj 94 rule is obta9able 9 a l5g? ( #aj 94 ,?is is effect$ by divid+ a #bj 94 7#ej cm47 scale l5g? into two p>ts & plac+ ^! on ! work+ edges ( ! rule & slide4 ,on ! upp] & l[] m>g9s ( ! face ( ! rule are ! two p>ts ( :at corresponds to ! ,a scale 9 ! ord9>y rule2 :ile 9 ! c5tre ( ! slide is ! scale ( log>i?ms :i*1 us$ 9 conjunction ) ! #ej cm4 scales on ! slide1 is virtually twice ! l5g? ( t ord9>ily obta9able 9 a #aj 94 rule4 ,! same rem>k applies to ! trigonometrical scales on ! "u face ( ! slide4 ,bo? ! s9e & tang5t scales are 9 two adjac5t l5g?s1 :ile on ! edge ( ! /ock ( ! rule1 bel[ ! cursor groove1 is a scale ( s9es ( small angles f #a #di to #e #dd4 ,?is is ref]r$ to ! #ej cm4 scales by an index projection on ! cursor4 ,if ,c & ,c are ! two p>ts ( ! scale on ! slide & ,d & ,d ! correspond+ scales on ! rule1 x is cle> t 9 multiply+ two factors #a on ,c c only be set directly to ! upp] scale ,d2 :ile #aj on ,c c only be set directly to ! l[] scale ,d4 ,h5ce if ! fir/ factor is great] ?an ab\t #cb1 ! cursor mu/ be us$ to br+ #a on ,c to ! fir/ factor on ,d4 ,simil>ly1 9 division1 num]ators & d5om9ators :i* occur on ,c & ,d or on ,c & ,d _c be plac$ 9 direct co9cid5ce b mu/ be set by ! aid ( ! cursor4 ,any unc]ta9ty 9 read+ ! result c be avoid$ by obs]v+ ! foll[+ rule3 ,if 9 sett+ ! index 7#a or #aj7 9 multiplication1 or 9 sett+ ! num]ator to ! d5om9ator 9 division1 x is necess>y to cross ! slide1 !n x w also be necess>y to cross ! slide to read ! product or quoti5t4 ,,! ,,univ]sal ,,slide ,,rule4--,9 ?is in/rum5t ! /ock c>ries two simil> scales runn+ f #a to #aj1 to :i* ! slide c be set4 ,above ! upp] "o is ! log>i?m scale & "u ! l[] "o ! scale ( squ>es #a to #ajj4 ,on ! edge ( ! /ock ( ! rule1 "u ! cursor groove1 is a scale runn+ f #a to #ajjj4 ,an index project+ f ! cursor enables ?is scale to be us$ ) ! scales on ! face ( ! rule1 giv+ cubes1 cube roots1 etc4 ,on ! slide1 ! l[] scale is an ord9>y scale1 #a to #aj4 ,! c5tre scale is ! fir/ "p ( a scale giv+ ! values ( s9 n cos n1 ?is scale be+ cont9u$ along ! upp] edge ( ! slide 7m>k$ 8s9-cos07 up to ! graduation #ej4 ,on ! rema9d] ( ?is l9e is a scale runn+ f "r to left 7#j to #ej7 & giv+ ! value ( cos#bn4 ,9 survey+1 ^! scales greatly facilitate ! calculations = ! horizontal di/ance betwe5 ! obs]v]'s /ation & any po9t1 & ! diff]5ce 9 height ( ^! two po9ts4 ,on ! back ( ! slide are scales = ! s9es & tang5ts ( angles4 ,! values ( ! s9es & tang5ts ( angles f #cd to #e #dd diff] little f "o ano!r1 & ! "o c5tre scale suffices = bo? functions ( ^! small angles4 ,,! ,,fix ,,slide ,,rule4--,?is is a /&>d rule 9 all respects1 except t ! ,a scale is displac$ by a di/ance 77_/7#d7 s t ov] #a on ,d is f\nd #jghed on ,a4 ,?is enables calculations relat+ to ! area & cubic cont5ts ( cyl9d]s to be det]m9$ v readily4 ,,! ,,begh9 ,,slide ,,rule4--,we h se5 t a disadvantage att5d+ ! use ( ! ord9>y ,c & ,d scales1 is t x is occasionally necess>y to trav]se ! slide ?r\gh its own l5g? 9 ord] to *ange ! indices or to br+ o!r p>ts ( ! slide into a readable position ) reg>d to ! /ock4 ,to obviate ?is disadvantage1 ,ts]epa*9sky devis$ an +5i\s arrangem5t :i* has s9ce be5 us$ 9 v>i\s rules1 notably 9 ! ,begh9 slide rule made by ,messrs4 ,tav]ni]-,gravt ( ,p>is4 ,9 ?is rule ! ,c & ,d scales are us$ z 9 ! /&>d rule1 b 9 place ( ! ,a & ,b scales1 we h ano!r pair ( ,c & ,d scales1 displac$ by "o-half ! l5g? ( ! rule4 ,! l[] pair ( scales may !re=e be reg>d$ z runn+ f #ajn to #ajn #a1 & ! upp] pair z runn+ f 7#aj7 #ajn to 7#aj7 #ajn #a4 ,) ?is arrangem5t1 )\t mov+ ! slide m ?an half its l5g?1 to ! left or "r1 x is always possible to comp>e all values betwe5 #a & #aj on ! two scales4 ,?is is a great advantage especially 9 cont9u\s work+4 ,ano!r comm5dable feature ( ! ,begh9 rule is ! pres5ce ( a rev]s$ ,c scale 9 ! c5tre ( ! slide1 ?us enabl+ su* calculations z a b c to be made ) "o sett+ ( ! slide4 ,on ! back ( ! slide are ?ree scales1 ! l[e/ ( :i*1 us$ ) ! ,d scale1 is a scale ( squ>es 7correspond+ to ! ord9>y ,b scale71 :ile on ! upp] edge is a scale ( s9es f #e #dd to #ij1 & 9 ! c5tre1 a scale ( tang5ts f #e #dc to #de4 ,on ! squ>e edge ( ! /ock1 "u ! cursor groove1 is ! log>i?m scale1 :ile on ! same edge1 above ! cursor groove1 are a s]ies ( gauge po9ts4 ,all ^! values are ref]r$ to ! face scales by index m>ks on ! cursor4 ,,! ,,&]son ,,slide ,,rule4--,! pr9ciple ( divid+ a long scale into sections z 9 ! ,precision rule1 has be5 ext5d$ 9 ! ,&]son slide rule made by ,messrs4 ,casella ,co41 ,london1 & %[n 9 ,fig4 #ag4 ,9 ?is ! slide c>ries a scale 9 f\r sections1 us$ 9 conjunction ) an exactly simil> set ( scale-l9es 9 ! upp] "p ( ! /ock4 ,on ! l[] "p ( ! /ock is a scale 9 eight sections giv+ ! squ>e roots ( ! upp] values4 ,9 ord] to set ! index ( ! slide to values 9 ! /ock1 two indices ( transp>5t celluloid are fix$ to ! slide ext5d+ ov] ! face ( ! rule z %[n 9 ! illu/ration4 ,z ea* scale section is #cj cm4 9 l5g?1 ! upp] l9es correspond to a s+le scale ( ne>ly #d ft41 & ! l[] set to "o ( ne>ly #h ft4 9 l5g?1 giv+ a correspond+ly l>ge increase 9 ! numb] ( subdivisions ( ^! scales1 & consequ5tly mu* great] accuracy4 ,9 ord] to decide ^u :i* l9e a result is to be f\nd1 sets ( 8l9e numb]s0 are m>k$ at ea* end ( ! rule & slide & also on ! metal frame ( ! cursor4 ,9 multiplication1 ! l9e numb] ( ! product is ! sum ( ! l9e numb]s ( ! factors if ! left index is us$1 or #a m ?an ?is sum if ! "r index is us$4 ,! illu/ration %[s ! multiplication ( #b by #d4 ,! left index is set to #b 7l9e numb]1 #a71 & ! cursor set to #d on ! slide 7l9e numb]1 #b72 h5ce1 z ! left index is us$1 ! result is f\nd on l9e ,no4 #c4 ,simil> rules are readily e/abli%$ = division4 ,! column ( l9e numb]s head$ #j is us$ = units1 t head$ #d = t5s1 & s on2 "o column is giv5 = t5?s1 head$ #d4 ,! squ>e root scale be>s simil> l9e numb]s1 s t ! squ>e root ( any value on ! upp] scales is f\nd on ! correspond+ly figur$ l9e bel[4 ,illu/ration3 ,,fig4 #ag4 ,,! ,,multiplex ,,slide ,,rule diff]s f ! ord9>y =m ( rule 9 ! arrangem5t ( ! ,b scale4 ,! "r-h& section ( ?is scale runs f left to "r z ord9>ily arrang$1 b ! left-h& section runs 9 ! rev]se direction1 & s furni%es a reciprocal scale4 ,at ! bottom ( ! groove1 "u ! slide1 !re is a scale runn+ f #a to #ajjj1 :i* is us$ 9 conjunction ) ! ,d scale1 read+s be+ ref]r$ !reto by a metal index on ! end ( ! slide4 ,by ?is means cubes1 cube roots1 etc41 c be read (f directly4 ,messrs4 ,eug5e ,dietzg5 ,co41 ,new ,york1 are ! mak]s4 ,,! 8,,long0 ,,slide ,,rule has "o scale 9 two sections along ! upp] & l[] p>ts ( ! /ock1 z 9 ! 8,precision0 rule4 ,! scale on ! slide is simil>ly divid$1 b ! graduations run 9 ! rev]se direction1 correspond+ to an inv]t$ slide4 ,h5ce ! rules = multiplication & division are ! rev]se ( ^? usually foll[$ 7page #cj74 ,on ! back ( ! slide is a s+le scale #a-aj1 & a scale #a-ajjj1 giv+ cubes ( ?is s+le scale4 ,by us+ ! fir/ 9 conjunction ) ! scales on ! /ock1 squ>es may be read1 :ile 9 conjunction ) ! cube scale1 v>i\s expressions involv+ squ>es1 cubes & _! roots may be evaluat$4 ,,hall',s ,,nautical ,,slide ,,rule consi/s ( two slides fitt+ 9 grooves 9 ! /ock1 & provid$ ) eight scales1 two on ea* slide1 & "o on ea* edge ( ea* groove4 ,:ile fulfill+ ! purposes ( an ord9>y slide rule1 x is ( especial s]vice to ! practical navigator 9 connection ) su* problems z ! 8r$uction ( an ex-m]idian sight0 & ! 8correction ( *ronomet] sights = error 9 latitude40 ,! rule1 :i* has _m o!r applications ( a simil> "*1 is made by ,mr4 ,j4 ,h4 ,/ew>d1 ,/r&1 ,london4 ,,long-,,scale ,,slide ,,rules ,x has be5 %[n t ! degree ( accuracy atta9able 9 slide-rule calculations dep5ds ^u ! l5g? ( scale employ$4 ,consid]ations ( g5]al conv5i5ce1 h[ev]1 r5d] simple /raight-scale rules ( m ?an #bj 94 9 l5g? inadmissible1 s t inv5tors ( long-scale slide rules1 9 ord] to obta9 a high degree ( precision1 comb9$ ) conv5i5ce 9 op]ation1 h be5 compell$ to modify ! arrangem5t ( scales usually employ$4 ,! pr9cipal me?ods adopt$ may be class$ "u ?ree v>ieties3 7#a7 ,! use ( a long scale 9 sectional l5g?s1 z 9 ,hannyngton's ,ext5d$ ,slide ,rule & ,?a*]'s ,calculat+ ,in/rum5t2 7#b7 ! employm5t ( a long scale laid 9 spiral =m ^u a disc1 z 9 ,fe>nley's ,univ]sal ,calculator & ,s*u]man's ,calculat+ ,in/rum5t2 & 7#c7 ! adoption ( a long scale w\nd helically ^u a cyl9d]1 ( :i* ,full]'s & ! 8,r4,h4,s40 ,calculat+ ,rules are examples4 ,,full]',s ,,calculat+ ,,rule4--,?is in/rum5t1 :i* is %[n 9 ,fig4 #ah1 consi/s ( a cyl9d] d capable ( be+ mov$ up & d[n & ar\nd ! cyl9drical /ock f1 :i* is held by ! h&le4 ,! log>i?mic scale-l9e is arrang$ 9 ! =m ( a helix ^u ! surface ( ! cyl9d] d1 & z x is equival5t to a /raight scale ( #ejj in*es1 or #da ft4 #h 941 x is possible to obta9 f\r1 & frequ5tly five1 figures 9 a result4 ,^u ref]5ce to ! figure x w be se5 t ?ree indices are employ$4 ,( ^!1 t lett]$ b is fix$ to ! h&le2 :ile two o!rs1 c & a 7^: di/ance ap>t is equal to ! axial l5g? ( ! complete helix71 are fix$ to ! inn]mo/ cyl9d] g4 ,?is latt] cyl9d] slides telescopically 9 ! /ock f1 enabl+ ! indices to be plac$ 9 any requir$ position relatively to d4 ,two o!r scales are provid$1 "o 7m7 at ! upp] end ( ! cyl9d] d1 & ! o!r 7n7 on ! movable index4 ,illu/ration3 ,,fig4 #ah4 ,9 us+ ! in/rum5t a giv5 numb] on d is set to ! fix$ index b1 & ei!r a or c is br\ght to ano!r numb] on ! scale4 ,?is e/abli%es a ratio1 & if ! cyl9d] is n[ mov$ s z to br+ any numb] to b1 ! f\r? t]m ( ! proportion w be f\nd "u a or c4 ,( c\rse1 9 multiplication1 "o factor is br\ght to b1 & a or c br\ght to #ajj4 ,! o!r factor is !n br\ght to a or c1 & ! result read (f "u b4 ,problems involv+ cont9u\s multiplication1 or comb9$ multiplication & division1 are v readily dealt )4 ,?us1 call+ ! fix$ index ,f1 ! upp] movable index ,a1 & ! l[] movable index ,b1 we h = a b c3--,br+ a to ,f2 ,a to #ajj2 b to ,a or ,b2 ,a to #ajj2 c to ,a or ,b & read ! product at ,f4 ,! maximum numb] ( figures 9 a product is ! sum ( ! numb] ( figures 9 ! factors & ?is results :5 all ! factors except ! fir/ h to be br\ght to ,b4 ,ea* "t a factor is br\ght to ,a1 #a is to be d$uct$ f t sum4 ,= division1 z a_/7m n71 br+ a to ,f2 ,a or ,b to m2 #ajj to ,a2 ,a or ,b to a2 #ajj to ,a & read ! quoti5t at ,f4 ,illu/ration3 ,,fig4 #ai4 ,! maximum numb] ( figures 9 ! quoti5t is ! diff]5ce betwe5 ! sum ( ! numb] ( figures 9 ! num]ator factors & ^? ( ! d5om9ator factors1 plus #a = ea* factor ( ! d5om9ator & ?is results :5 ,a has to be set to all ! factors ( ! d5om9ator & all ! factors ( ! num]ator except ! fir/ br\ght to ,b4 ,ea* "t ,b is set to a d5om9ator factor or a num]ator factor is br\ght to ,a1 #a is to be d$uct$4 ,log>i?ms ( numb]s are obta9$ by us+ ! scales m & n & h5ce p[]s & roots ( any magnitude may be obta9$ by ! proc$ure already fully expla9$4 ,! in/rum5t illu/rat$ is made by ,messrs4 ,w4 ,f4 ,/anley ,co41 ,limit$1 ,london4 ,,! 8,r4,h4,s40 ,,calculator4--,9 ?is calculator1 design$ by ,pr(4 ,r4 ,h4 ,smi?1 ! scale-l9e1 :i* is #ej 94 long1 is also arrang$ 9 a spiral =m 7,fig4 #ai71 b 9 ?is case x is wrapp$ ar\nd ! c5tral portion ( a tube :i* is ab\t 94 9 diamet] & #i;94 long4 ,a slott$ hold]1 capable ( slid+ ^u ! pla9 portions ( ?is tube1 is provid$ ) f\r horns1 ^! be+ =m$ at ! ends ( ! two wide op5+s ?r\gh :i* ! scale is read4 ,an out] r+ c>ry+ two horns completes ! arrangem5t4 ,"o ( ! horns ( ! hold] be+ plac$ 9 agreem5t ) ! fir/ factor1 & "o ( ! horns ( ! r+ ) ! second factor1 ! hold] is mov$ until ! ?ird factor falls "u ! same horn ( ! r+1 :5 ! result+ f\r? t]m w be f\nd "u ! same 7"r or left7 horn ( ! hold]1 at ei!r end ( ! slot4 ,9 multiplication1 #ajj or #ajjj is tak5 = ! second factor 9 ! above proportion1 z already expla9$ 9 connection ) ,full]'s rule2 inde$1 g5]ally1 ! mode ( op]ation is ess5tially simil> to t foll[$ ) ! =m] in/rum5t4 ,! scale %[n on "o edge ( ! op5+ 9 ! hold]1 toge!r ) ! circul> scale at ! top ( ! spiral1 enables ! mantiss ( log>i?ms ( numb]s to be obta9$1 & ?us problems involv+ p[]s & roots may be dealt ) q readily4 ,?is in/rum5t is suppli$ by ,mr4 ,j4 ,h4 ,/ew>d1 ,london4 ,,?a*]',s ,,calculat+ ,,in/rum5t1 %[n 9 ,fig4 #bj1 consi/s ( a cyl9d] #d 94 9 diamet] & #ah 94 long1 :i* c be giv5 bo? a rot>y & a longitud9al movem5t )9 an op5 framework compos$ ( tw5ty triangul> b>s4 ,^! b>s are connect$ to r+s at _! ends1 :i* c be rotat$ 9 /&>ds fix$ to ! basebo>d4 ,! scale on ! cyl9d] consi/s ( =ty sectional l5g?s1 b ( ea* scale-l9e t "p :i* appe>s on ! "r-h& half ( ! cyl9d] is repeat$ on ! left-h& half1 "o l9e 9 advance4 ,h5ce ea* half ( ! cyl9d] virtually conta9s two complete scales foll[+ r\nd 9 regul> ord]4 ,on ! l[] l9es ( ! triangul> b>s are scales exactly correspond+ to ^? on ! cyl9d]1 :ile ^u ! upp] l9es ( ! b>s & n 9 contact ) ! slide is a scale ( squ>e roots4 ,illu/ration3 ,,fig4 #bj4 ,by rotat+ ! slide any l9e on x may be br\ght opposite any l9e 9 frame & by a longitud9al movem5t any graduation on ^! l9es may be br\ght into agreem5t4 ,! :ole c be rotat$ 9 ! support+ /&>ds 9 ord] to br+ any read+ into view4 ,z %[n 9 ! illu/ration1 a magnifi] is provid$1 ?is be+ conv5i5tly m\nt$ on a b>1 along :i* x c be mov$ z requir$4 ,,sectional ,,l5g? ,,or ,,gridiron ,,slide ,,rules4--,! idea ( break+ up a long scale into sectional l5g?s is due to ,dr4 ,j4 ,d4 ,ev]ett1 :o describ$ su* a gridiron type ( slide rule 9 #ahff4 ,hannyngton's ,ext5d$ ,slide ,rule is on ! same pr9ciple4 ,bo? in/rum5ts h ! l[] scale repeat$4 ,h4 ,*]ry 7#ahhj7 appe>s to h be5 ! fir/ to %[ t su* duplication c\ld be avoid$ by provid+ two fix$ index po9ts 9 addition to ! natural indices ( ! scale4 ,^! additional indices are %[n at #aj & #ajj 9 ,fig4 #ba1 :i* repres5ts ! l[] %eet ( ,*]ry's ,calculator on a r$uc$ scale4 ,! upp] memb] ( ! calculator consi/s ( a transp>5t %eet rul$ ) p>allel l9es1 :i* co9cide ) ! l9es ( ! l[] scale :5 ! indices ( bo? are plac$ 9 agreem5t4 ,to multiply "o numb] by ano!r1 "o ( ! indices on ! upp] %eet is plac$ to "o ( ! factors1 & ! position ( :i*ev] index falls "u ! transp>5t %eet is not$ on ! latt]4 ,br++ ! latt] po9t to ! o!r factor1 ! result is f\nd "u :i*ev] index lies on ! c>d4 ,9 o!r arrangem5ts ! inv5tor us$ transp>5t scales1 ! graduations runn+ 9 a rev]se direction to ^? ( ! l[] scale4 ,9 ?is case1 a factor on ! upp] scale is set to ! o!r factor on ! l[]1 & ! result read at ! available index4 ,illu/ration3 ,,fig4 #ba4 ,,proell',s ,,pocket ,,calculator is an application ( ! la/-nam$ pr9ciple4 ,x comprises a l[] c>d arrang$ z ,fig4 #ba1 ) an upp] %eet ( transp>5t celluloid on :i* is a simil> scale runn+ 9 ! rev]se direction4 ,= cont9u$ multiplication & division1 a ne$le 7suppli$ ) ! in/rum5t7 is us$ z a sub/itute = a cursor1 to fix ! position ( ! int]m$iate results4 ,a s]ies ( index po9ts on ! l[] c>d enable squ>e & cube roots to be extract$ v readily4 ,?is calculator is suppli$ by ,messrs4 ,john ,j4 ,griff9 ,sons1 ,ltd41 ,london4 ,,circul> ,,calculators4 ,al?\gh ! #aj 94 slide rule is probably ! mo/ s]viceable =m ( calculat+ in/rum5t = g5]al purposes1 _m pref] ! m portable circul> calculator1 ( :i* _m v>ieties h be5 introduc$ dur+ rec5t ye>s4 ,! advantages ( ?is type are3 ,x is m compact & conv5i5tly c>ri$ 9 ! wai/coat pocket4 ,! scales are cont9u\s1 s t no trav]s+ ( ! slide f #a to #aj is requir$4 ,! dial c be set quickly to any value2 !re is no tr\ble ) tight or ill-fitt+ slides4 ,! disadvantages ( mo/ =ms are3 ,_m problems involve m op]ations ?an a /raight rule4 ,! results be+ read "u f+]s or po9t]s1 an error due to p>allax is introduc$1 s t ! results g5]ally are n s accurate z ) a /raight rule4 ,! inn] scales are %ort1 & !re=e are read ) less accuracy4 ,special scale circles are ne$$ = cubes & cube roots4 ,! slide _c be rev]s$ or inv]t$4 ,illu/ration3 ,,fig4 #bb4 ,illu/ration3 ,,fig4 #bc4 ,,! ,,b\*] ,,calculator4--,?is circul> calculator resembles a /em-w9d+ wat*1 be+ ab\t #b 94 9 diamet] & 94 9 ?ickness4 ,! in/rum5t has two dials1 ! back "o be+ fix$1 :ile ! front "o1 ,fig4 #bb 7%[+ ! =m made by ,messrs4 ,w4 ,f4 ,/anley1 ,london71 turns ^u ! l>ge c5tre arbor %[n4 ,?is movem5t is effect$ by turn+ ! mill$ head ( ! /em-w9d]4 ,! small c5tre axis1 :i* is turn$ by rotat+ ! mill$ head at ! side ( ! case1 c>ries two f9e ne$le po9t]s1 "o mov+ ov] ea* dial1 & s fix$ on ! axis t "o po9t] always lies ev5ly ov] ! o!r4 ,a f9e index or po9t] fix$ to ! case 9 l9e ) ! axis ( ! w9d+ /em1 ext5ds ov] ! f\r scales ( ! movable dial z %[n4 ,( ^! scales1 ! second f ! out] is ! ord9>y log>i?mic scale1 :i* 9 ?is in/rum5t corresponds to a /raight scale ( ab\t #d;94 9 l5g?4 ,! two inn] circles give ! squ>e roots ( ! numb]s on ! prim>y log>i?mic scale1 ! small] circle conta9+ ! squ>e roots ( values betwe5 #a & #cafb 7 7#aj771 :ile ! o!r section corresponds to values betwe5 #cafb & #aj4 ,! out] circle is a scale ( log>i?ms ( s9es ( angles1 ! correspond+ s9es ( :i* c be read (f on ! ord9>y scale4 ,on ! fix$ or back dial !re are also f\r scales1 ^! be+ arrang$ z 9 ,fig4 #bc4 ,! out] ( ^! is a scale ( equal p>ts1 :ile ! ?ree inn] scales are sep>ate sections ( a scale giv+ ! cube roots ( ! numb]s tak5 on ! ord9>y log>i?mic scale & ref]r$ !reto by means ( ! po9t]s4 ,9 divid+ ?is cube-root scale into sections1 ! same me?od is adopt$ z 9 ! case ( ! squ>e-root scale4 ,?us1 ! smalle/ circle conta9s ! cube roots ( numb]s betwe5 #a & #aj1 & is !re=e graduat$ f #a to #baed2 ! second circle conta9s ! cube roots ( numb]s betwe5 #aj & #ajj1 be+ graduat$ f #baed to #dfeg2 :ile ! ?ird section1 9 :i* are f\nd ! cube roots ( numb]s betwe5 #ajj & #ajjj1 c>ries ! graduations f #dfeg to #aj4 ,:at has be5 said 9 an e>li] section reg>d+ ! notation ( ! slide rule may 9 g5]al be tak5 to apply to ! scales ( ! ,b\*] calculator4 ,! mann] ( us+ ! in/rum5t is1 h[ev]1 n q s evid5t1 al?\gh f :at foll[s x w be se5 t ! op]ative pr9ciple--t ( v>i\sly comb9+ l5g?s ( a log>i?mic scale--is ess5tially simil>4 ,9 ?is case1 h[ev]1 x is se5 t 9 place ( ! /raight scale-l5g?s %[n 9 ,fig4 #d1 we require to add or subtract arc-l5g?s ( ! circul> scales1 :ile1 fur!r1 x is evid5t t 9 ! abs5ce ( a fix$ scale 7correspond+ to ! /ock ( ! slide rule7 ^! op]ations _c be directly p]=m$ z 9 ! ord9>y =m ( in/rum5t4 ,h[ev]1 by ! aid ( ! fix$ index & ! movable po9t]1 we c effect ! desir$ comb9ation ( ! scale-l5g?s 9 ! foll[+ mann]4 ,assum+ x is desir$ to multiply #b by #c1 ! dial is turn$ 9 a backw>d direction until #b on ! ord9>y scale lies "u ! fix$ index1 aft] :i* ! movable po9t] is set to #a on ! scale4 ,z n[ set1 x is cle> t ! arc-l5g? #a-b is spac$ (f betwe5 ! fix$ index & ! movable po9t]1 & x n[ only rema9s to add to ?is def9ite arc-l5g? a fur!r l5g? ( #a-c4 ,to d ?is we turn ! dial /ill fur!r backw>d until ! arc #a-c has pass$ "u ! movable po9t]1 :5 ! result1 #f1 is read "u ! fix$ index4 ,a little consid]ation w %[ t any o!r scale l5g? may be add$ to t includ$ betwe5 ! fix$ & movable po9t]s1 or1 9 o!r words1 any numb] on ! scale may be multipli$ by #b by br++ ! numb] to ! movable po9t] & read+ ! result "u ! fix$ index4 ,! rule = multiplication is n[ evid5t4 ,rule = ,multiplication4--,set "o factor to ! fix$ index & br+ ! po9t] to #a on ! scale2 set ! o!r factor to ! po9t] & read ! result "u ! fix$ index4 ,) ! explanation j giv51 ! process ( division ne$s little explanation4 ,x is cle> t to divide #f by #c1 an arc-l5g? #a-c is to be tak5 f a l5g? #a-f4 ,to ?is end we set #f to ! index 7correspond+ 9 effect to pass+ a l5g? #a-f to ! left ( t ref]5ce po9t7 & set ! po9t] to ! divisor #c4 ,z n[ set1 ! arc #a-f is includ$ betwe5 #a on ! scale & ! index1 :ile ! arc #a-c is includ$ betwe5 #a on ! scale & ! po9t]4 ,obvi\sly if ! dial is n[ turn$ =w>d until #a on ! scale agrees ) ! po9t]1 an arc #a-c w h be5 d$uct$ f ! l>g] arc #a-f1 & ! rema9d]1 repres5t+ ! result ( ?is op]ation1 w be read "u ! index z #b4 ,rule = ,division4--,set ! divid5d to ! fix$ index1 & ! po9t] to ! divisor2 turn ! dial until #a on ! scale agrees ) ! po9t]1 & read ! result "u ! fix$ index4 ,! =ego+ me?od be+ an inv]sion ( ! rule = multiplication1 is easily rememb]$ & is g5]ally advis$4 ,ano!r plan is1 h[ev]1 pref]able :5 a s]ies ( divisions are to be effect$ ) a con/ant divisor--i4e41 :5 b 9 7a7_/7b7 x is con/ant4 ,9 ?is case #a on ! scale is set to ! index & ! po9t] set to b2 !n if any value ( a is br\ght to ! po9t]1 ! quoti5t x w be f\nd "u ! index4 ,comb9$ ,multiplication & ,division1 z 7a b c7_/7m n7 x1 c be readily p]=m$1 :ile cases ( cont9u$ multiplication evid5tly come "u ! same category1 s9ce a b c 7a b c7_/7#a #a7 x4 ,su* cases z a_/7m n r7 x are reg>d$ z 7a #a #a #a7_/7m n r7 x2 :ile 7a b c7_/7m7 x is simil>ly modifi$1 tak+ ! =m 7a b c7_/7m #a7 x4 ,9 all cases ! expression mu/ be arrang$ s t !re is "o m factor 9 ! num]ator ?an 9 ! d5om9ator1 #a's be+ introduc$ z (t5 z requir$4 ,! simple op]ations ( multiplication & division involve a simil> disposition ( factors1 s9ce f ! rules giv5 x is evid5t t m n is actually reg>d$ z 7m n7_/7#a71 :ile 7m7_/7n7 becomes 9 effect 7m #a7_/7n74 ,x is important to note ! g5]al applicability ( ?is arrangem5t-rule1 z x w be f\nd ( great assi/ance 9 solv+ m complicat$ expressions4 ,z ) ! ord9>y =m ( slide rule1 ! factors 9 su* an expression z 7a b c7_/7m n7 x are tak5 9 ! ord]3--#a/ factor ( num]ator2 #a/ factor ( d5om9ator2 #bnd factor ( num]ator2 #bnd factor ( d5om9ator1 & s on2 ! #a/ factor z a be+ set to ! index1 & ! result x be+ f9ally read at ! same po9t ( ref]5ce4 ,,ex4--7#ci #adb #fc7_/7acg #ai7 #acd4 ,comm5ce by sett+ #ci to ! index1 & ! po9t] to #acg2 br+ #adb to ! po9t]2 po9t] to #ai2 #fc to ! po9t]1 & read ! result #acd at ! index4 ,x %\ld be not$ t aft] ! fir/ factor is set to ! fix$ index1 ! po9t] is set to ea* ( ! divid+ factors z !y ent] into ! calculation1 :ile ! dial is mov$ = ea* ( ! multiply+ factors4 ,?us ! dial is fir/ mov$ 7sett+ ! fir/ factor to ! index71 !n ! po9t]1 !n ! dial1 & s on4 ,numb] ( ,digits 9 ! ,result4--,if rules are pref]r$ to ! plan ( r\ghly e/imat+ ! result1 ! g5]al rules giv5 on pages #ba & #be %\ld be employ$ = simple cases ( multiplication & division4 ,= comb9$ multiplication & division1 modify ! expression1 if necess>y1 by introduc+ #a's1 z already expla9$1 & subtract ! sum ( ! d5om9ator digits f ! sum ( num]ator digits4 ,!n proce$ by ! au?or's rule1 z foll[s3-- ,always turn dial to ! ,,left2 i4e41 aga9/ ! h&s ( a wat*4 ,note dial movem5ts only2 ignore ^? ( ! po9t]4 ,ea* "t #a on dial agrees ) or passes fix$ index1 ,,add #a to ! above diff]5ce ( digits4 ,ea* "t #a on dial agrees ) or passes po9t]1 ,,d$uct #a f ! above diff]5ce ( digits4 ,treat cont9u$ multiplication 9 ! same way1 c\nt+ ! #a's us$ z d5om9ator digits z "o less ?an ! numb] ( multipli$ factors4 ,,ex4--7#hf #jgc #ajb7_/7ce #jbc7 #gie #giedgc4 ,set #hf to index & po9t] to #ce4 ,br+ #jgc to po9t] 7not+ t #a on ! scale passes ! index7 & set po9t] to #jbc4 ,set #ajb to po9t] 7not+ t #a on ! scale passes ! po9t]7 & read "u index #gie4 ,!re are #a #j #a #b num]ator digits & #a #j #a d5om9ator digit2 :ile #a is to be add$ & #a d$uct$ z p] rule4 ,b z ! latt] cancel1 ! digits 9 ! result w be #b #a #a4 ,:5 mov+ ! dial to ! left w cause #a on ! dial to pass bo? index & po9t] 7?us cancell+71 ! dial may be turn$ back to make ! sett+4 ,x w be und]/ood t :5 #a is ! fir/ num]ator1 & #a on ! dial is !re=e set to ! index1 no digit addition w be made = ?is1 z ! actual op]ation ( calculat+ has n be5 comm5c$4 ,9 ! ,/anley-,b\*] calculator 7,fig4 #bc7 a small c5tre scale is add$1 on :i* a f+] indicates automatically ! numb] ( digits to be add$ or d$uct$2 ! me?od ( calculat+1 h[ev]1 diff]s f ! =ego+4 ,to avoid turn+ back to #j at ! comm5cem5t ( ea* calculation1 a circle is gr\nd on ! glass face1 s t a p5cil m>k c be made !reon to %[ ! position ( ! f+] :5 comm5c+ a calculation4 ,to ,f9d ! ,squ>e ( a ,numb]4-- ,set ! numb]1 on "o or o!r ( ! squ>e root scales1 to ! index1 & read ! requir$ squ>e on ! ord9>y scale4 ,to ,f9d ! ,squ>e ,root ( a ,numb]4--,set ! numb] to ! index1 & if !re is an odd numb] ( digits 9 ! numb]1 read ! root on ! inn] circle2 if an ev5 numb]1 on ! second circle4 ,to ,f9d ! ,cube ( a ,numb]4--,set #a on ! ord9>y scale to ! index1 & ! po9t] 7on ! back dial7 to ! numb] on "o ( ! ?ree cube-root scales4 ,!n "u ! po9t] read ! cube on ! ord9>y scale4 ,to ,f9d ! ,cube ,root ( a ,numb]4--,set #a to index1 & po9t] to numb]4 ,!n read ! cube root "u ! po9t] on "o ( ! ?ree inn] circles on ! back dial4 ,if ! numb] has #a1 #d1 #g1 #aj or #b1 #e1 etc41 digits1 use ! inn] circle4 #b1 #e1 #h1 #aa or #a1 #d1 etc41 second circle4 #c1 #f1 #i1 #ab or #j1 #c1 etc41 ?ird circle4 ,= ,p[]s or ,roots ( ,high] ,d5om9ation4--,set #a to index1 ! po9t] to ! numb] on ! ord9>y scale1 & read on ! out] circle on ! back dial ! mantissa ( ! log>i?m4 ,add ! *>act]i/ic 7see p4 #df71 multiply by ! p[] or divide by ! root1 & set ! po9t] to ! mantissa ( ! result on ?is out] circle4 ,"u ! po9t] on ! ord9>y scale read ! numb]1 obta9+ ! numb] ( figures f ! *>act]i/ic4 ,to ,f9d ! ,s9es ( ,angles4--,set #a to index1 po9t] to ! angle on ! out] circle1 & read "u ! po9t] ! natural s9e on ! ord9>y scale2 also "u ! po9t] on ! out] circle ( ! back dial read ! log>i?mic s9e4 ,,! ,,hald5 ,,calculex4--,aft] ! introduction ( ! ,b\*] calculator 9 #ahgf1 circul> in/rum5ts1 su* z ! ,*>p5ti] calculator1 7 introduc$1 9 :i* a disc turn$ )9 a fix$ r+1 s t scales on ! faces ( bo? c\ld be set toge!r & ratios e/abli%$ z on ! slide rule4 ,cultriss's ,calculat+ ,disc is ano!r in/rum5t on ! same pr9ciple4 ,! ,hald5 ,calculex1 ( :i* half-size illu/rations are giv5 9 ,figs4 #bd & #be1 repres5ts a consid]able improvem5t ^u ^! e>ly in/rum5ts4 ,x consi/s ( an out] metal r+ c>ry+ a fix$-scale r+1 )9 :i* is a dial4 ,on ea* side ( ?is dial are flat mill$ heads1 s t by hold+ ^! betwe5 ! ?umb & =ef+] ! dial c be set quickly & conv5i5tly4 ,! protect+ glass discs1 :i* are n fix$ 9 ! metal r+ b are arrang$ to turn !re91 c>ry f9e cursor l9es1 & z ^! are on ! side next to ! scales a v close sett+ c be made q free f ! effects ( p>allax4 ,?is con/ruction n only avoids ! use ( me*anism1 ) its risk ( d]angem5t1 b r$uces ! bulk ( ! in/rum5t v consid]ably1 ! ?ickness be+ ab\t 94 ,on ! front face1 ,fig4 #bd1 ! fix$ r+ c>ries an out] ev5ly-divid$ scale1 giv+ log>i?ms1 & an ord9>y scale1 #a-aj1 :i* works 9 conjunction ) a simil> scale on ! edge ( ! dial4 ,! two inn] circles give ! squ>e roots ( values on ! ma9 scales z 9 ! ,b\*] calculator4 ,on ! back face1 ,fig4 #be1 ! r+ be>s an out] scale1 giv+ s9es ( angles f #f to #ij & an ord9>y scale1 #a-aj1 z on ! front face4 ,! scales on ! dial are all rev]s$ 9 direction 7runn+ f "r to left71 ! out] "o consi/+ ( an ord9>y 7b inv]se7 scale1 #a-aj1 :ile ! ?ree inn] circles give ! cube roots ( values on ?is inv]se scale4 ,z ! f9e cursor l9es ext5d ov] all ! scales1 a v>iety ( calculations c be effect$ v readily & accurately4 ,illu/ration3 ,,fig4 #bd4 ,illu/ration3 ,,fig4 #be4 ,,sp]ry',s ,,pocket ,,calculator1 made by ! ,keuffel & ,ess] ,company1 ,new ,york 7,fig4 #bf71 has two rotat+ dials1 ea* ) its own po9t] & fix$ index4 ,! ,s dial has an out] scale ( equal p>ts1 an ord9>y log>i?mic scale1 & a squ>e-root scale4 ,! ,l dial has a s+le log>i?mic scale arrang$ spirally1 9 ?ree sections1 giv+ a scale l5g? ( #ab;94 ,! po9t]s are turn$ by ! small mill$ head1 :i* is conc5tric ) ! mill$ ?umb-nut by :i* ! two dials are rotat$4 ,! ge>+ is su* t bo? ! ,l dial & its po9t] rotate ?ree times z fa/ z ! ,s dial & po9t]4 ,all ! usual calculations c be made ) ! spiral scale1 z ) ! ,b\*] calculator1 & ! result read (f on "o or o!r ( ! ?ree scale-sections4 ,frequ5tly ! po9t at :i* to read ! result is obvi\s1 b o!rwise a ref]5ce to ! s+le scale on ! ,s dial w %[ on :i* ( ! ?ree spirals ! result is to be f\nd4 ,illu/ration3 ,,fig4 #bf4 ,! ,k & ,e ,calculator1 also made by ! ,keuffel & ,ess] ,company1 is %[n 9 ,figs4 #bg & #bh4 ,x has two dials1 ( :i* only "o revolves4 ,?is1 z %[n 9 ,fig4 #bg1 has an ord9>y log>i?mic scale & a scale ( squ>es4 ,!re is an index l9e engrav$ on ! glass ( ! in/rum5t4 ,! fix$ dial has a scale ( tang5ts1 a scale ( equal p>ts & a scale ( s9es1 ! latt] be+ on a two-turn spiral4 ,! po9t]s1 :i* move toge!r1 are turn$ by a mill$ nut & ! movable dial by a ?umb-nut1 z 9 ,sp]ry's ,calculator1 ,fig4 #bf4 ,illu/ration3 ,,fig4 #bg4 ,illu/ration3 ,,fig4 #bh4 ,,slide ,,rules ,,= ,,special ,,calculations4 ,,eng9e ,,p[] ,,comput]4--,a typical example ( special slide rules is %[n 9 ,fig4 #bi1 :i* repres5ts1 on a scale ( ab\t half full size1 ! au?or's ,p[] ,comput] = ,/eam1 ,gas1 & ,oil ,eng9es4 ,?is1 z w be se51 consi/s ( a /ock1 on ! l[] portion ( :i* is a scale ( cyl9d] diamet]s1 :ile ! upp] portion c>ries a scale ( horse-p[]s4 ,9 ! groove betwe5 ^! scales are two slides1 also c>ry+ scales1 & capable ( slid+ 9 edge contact ) ! /ock & ) ea* o!r4 ,?is in/rum5t gives directly ! brake horse-p[] ( any /eam1 gas1 or oil eng9e2 ! indicat$ horse-p[]1 ! dim5sions ( an eng9e to develop a giv5 p[]1 & ! me*anical effici5cy ( an eng9e4 ,! calculation ( pi/on spe$1 velocity ratios ( pulleys & ge> :eels1 ! circumf]5tial spe$ ( pulleys1 & ! velocity ( belts & ropes driv5 !reby1 are among ! o!r pr9cipal purposes = :i* ! comput] may be employ$4 ,illu/ration3 ,,fig4 #bi4 ,,! ,,smi?-,,davis ,,piecework ,,balance ,,calculator has two scales1 #aa feet long1 hav+ a range f #a;d4 to #bj1 & m>k$ s t !y c be us$ ei!r = money or "t calculations4 ,! scales are plac$ on ! rims ( two simil> :eels & s arrang$ t ! divid$ edges come toge!r4 ,! :eels are m\nt$ on a sp9dle c>ri$ at ea* end 9 ! be>+s ( a support+ /&4 ,! :eels are press$ toge!r by a spr+1 & move z "o4 ,to set ! scales "o to ! o!r1 a treadle ge> is arrang$ to take ! pressure ( ! spr+ s t :5 ! fix$ :eel is held by ! left h& ! free :eel c be rotat$ by ! "r h& 9 ei!r direction4 ,:5 ! am\nt ( ! balance has be5 set to ! comb9$ weekly wage ! treadle is releas$ lock+ ! two :eels toge!r1 :5 ! :ole c be turn$ & ! am\nts respectively due to ea* man read (f opposite 8 weekly wage4 ,! ,smi?-,davis ,premium ,calculator is on ! same pr9ciple b ! scales are ab\t #d feet #f in*es long & ! :eels spr+-controll$4 ,bo? in/rum5ts are suppli$ by ,messrs4 ,john ,davis ,son1 ,ltd41 ,d]by4 ,,! ,,ba9es ,,slide ,,rule4--,9 ?is rule1 inv5t$ by ,mr4 ,h4 ,m4 ,ba9es1 ,lahore1 f\r slides c>ry+ scales are arrang$ to move1 ea* 9 edge contact ) ! next4 ,! slides are kept 9 contact & giv5 ! desir$ relative movem5t "o to ! o!r1 by be+ atta*$ 7at ! back71 to a jo9t$ p>allelogram4 ,on ?is pr9ciple :i* is ( g5]al application1 ! inv5tor has made a rule = ! solution ( problems cov]$ by ,flamant's =mula = ! fl[ ( wat] 9 ca/-iron pipes3--,v #gfbh;ds1 9 :i* s is ! s9e ( ! incl9ation or loss ( head2 d ! diamet] ( ! pipe 9 in*es & ,v ! velocity 9 feet p] second4 ,! =mula ,q ,,av is also includ$ 9 ! scope ( ! rule1 ,q be+ ! dis*>ge 9 cubic feet p] second & ,a ! cross sectional area ( ! pipe 9 squ>e in*es4 ,,f>m>',s ,,pr(it-,,calculat+ ,,rule4--,! application ( ! slide rule to comm]cial calculations has be5 (t5 attempt$1 b ! degree ( accuracy requir$ necessitates ! use ( a long scale1 & g5]ally ?is results 9 a cumb]some in/rum5t4 ,9 ,f>m>'s ,pr(it-calculat+ ,rule ! money scale is arrang$ 9 t5 sections1 ^! be+ m\nt$ 9 p>allel =m on a roll] :i* takes ! place ( ! upp] scale ( an ord9>y rule4 ,! roll]1 :i* is 94 9 diamet]1 is c>ri$ 9 brackets secur$ to ea* end ( ! /ock1 s t by rotat+ ! roll] any section ( ! money scale c be br\ght into read+ ) ! scale on ! upp] edge ( ! slide & ) :i* ! roll] is 9 contact4 ,?is scale gives p]c5tages1 & enables calculations to be made %[+ pr(it on turnov]1 pr(it on co/1 & disc\nt4 ,! l[] scale on ! slide1 & t on ! /ock adjac5t to x1 are simil> to ! ,a & ,b scales ( an ord9>y rule4 ,! in/rum5t is suppli$ by ,messrs4 ,j4 ,cas>telli ,son1 ,man*e/]4 ,,con/ructional ,,improvem5ts ,,9 ,,slide ,,rules4 ,! att5tion ( in/rum5t mak]s is n[ be+ giv5 to ! devis+ ( means = ensur+ ! smoo? & ev5 work+ ( ! slide 9 ! /ock ( ! rule4 ,9 "s cases v good results are obta9$ by slitt+ ! back ( ! /ock to give m ela/icity4 ,9 ! rules made by ,messrs4 ,john ,davis ,son1 a metal /rip1 slightly curv$ 9 cross section z %[n at ,a 7,fig4 #cj71 runs = ! full l5g? ( ! /ock to :i* x is fa/5$ at int]vals4 ,ne> ea* end ( ! rule1 op5+s ab\t #a 94 long are made 9 ! metal back+ ?r\gh :i* ! scales on ! back ( ! slide c be read4 ,to prev5t w>p+ "u v>y+ climatic conditions bo? ! /ock ( ! rule & ! slide are ( composite con/ruction4 ,! base ( ! /ock is ( mahogany1 :ile ! groov$ sides1 firmly secur$ to ! base1 are ( boxwood4 ,simil>ly ! c5tre portion ( ! slide is ( mahogany & ! tongu$ sides ( boxwood4 ,celluloid also ent]s into ! con/ruction1 a /rip ( ?is mat]ial be+ laid along ! bottom ( ! groove 9 ! /ock4 ,a f9e groove runs along ! c5tre ( ?is /rip 9 ord] to give ela/icity & to all[ ! sides ( ! /ock to be press$ toge!r slightly to adju/ ! fitt+ ( ! slide4 ,z a fur!r means ( adju/m5t ! mak]s fit metal clips at ea* end ( ! rule1 s t by tight5+ two small screws ! /ock c be clos$ on ! slide :5 necess>y4 ,illu/ration3 ,,fig4 #cj4 ,illu/ration3 ,,fig4 #cb4 ,illu/ration3 ,,fig4 #ca4 ,9 ! rule made by ! ,keuffel & ,ess] ,company ( ,new ,york1 "o /rip is made adju/able 7,fig4 #cb74 ,,! ,,accuracy ,,( ,,slide ,,rule ,,results4 ,! degree ( accuracy obta9able ) ! slide rule dep5ds prim>ily ^u ! l5g? ( ! scale employ$1 b ! accuracy ( ! graduations1 ! eyesight ( ! op]ator1 &1 9 p>ticul>1 8 ability to e/imate int]polat$ values1 are all factors :i* affect ! result4 ,us+ ! l[] scales & work+ c>efully ! error %\ld n greatly exce$ #jae p] c5t4 ) %ort calculations4 ,) successive sett+s1 ! discrepancy ne$ n necess>ily be great]1 z ! errors may be neutralis$2 b ) rapid work+ ! p]c5tage error may be d\bl$4 ,h[ev]1 mu* dep5ds ^u ! graduation ( ! scales4 ,rules 9 :i* "o or m ( ! indices h be5 ?ick5$ to conceal "s slight inaccuracy %\ld be avoid$4 ,! l9e on ! cursor %\ld be %>p & f9e & bo? slide & cursor %\ld move smoo?ly or good "w _c be done4 ,occasionally a little vasel9e or clean tall[ %\ld be appli$ to ! edges ( ! slide & cursor4 ,t ! p]c5tage error is con/ant ?r\gh\t ! scale is se5 by sett+ #a on ,c to #aja on ,d1 :5 "u #b is #bjb2 "u #c1 #cjc2 "u #e1 #eje1 etc41 ! sev]al read+s %[+ a uni=m error ( #a p] c5t4 ,a me?od ( obta9+ a clos] read+ ( a fir/ sett+ or ( a result on ,d has be5 sugge/$ to ! au?or by ,mr4 ,m4 ,a9slie1 ,b4,sc4 ,if any graduation1 z #d on ,c1 is set to #c on ,d1 x is se5 t #d ma9 divisions on ,c 7#dj-dd7 are equal 9 scale l5g? to #c ma9 divisions on ,d 7#cj-cc74 ,h5ce1 v approximately1 #a division on ,c is equal to #jge ( a division on ,d1 ?is ratio be+ %[n1 ( c\rse1 on ,d "u #aj on ,c4 ,suppose 7#dc7 to be requir$4 ,sett+ ! cursor to #dc on ,a1 x is se5 t ! root is some?+ m ?an #bjf4 ,move ! slide until a ma9 division is f\nd on ,c1 :i* exactly corresponds to ! int]val betwe5 #b & ! cursor l9e1 on ,d4 ,! division #bg-bh j fits1 giv+ a read+ "u #aj on ,c1 ( #gd4 ,h5ce ! root is read z #bjgd4 ,= ! high] p>ts ( ! scale1 ! subdivisions1 #a-aa1 etc41 are us$ 9 place ( ma9 divisions4 ,! me?od is probably m int]e/+ ?an useful1 s9ce 9 mo/ op]ations ! inaccuracies introduc$ 9 mak+ sett+s w impose a limit on ! reliable figures ( ! result4 ,= ! majority ( eng9e]+ calculations1 ! slide rule w give an accuracy consi/5t ) ! accuracy ( ! data usually available4 ,= "s purposes1 h[ev]1 log>i?mic section pap] 7! use ( :i* ! au?or has advocat$ = ! la/ tw5ty ye>s7 w be f\nd especially useful1 m p>ticul>ly 9 calculations involv+ expon5tial =mul4 ,,app5dix4 ,,new ,,slide ,,rules--,,fif? ,,roots1 ,,etc4--,,! ,,solution ,,( ,,algebraic ,,equations--,,gauge ,,po9ts ,,& ,,signs ,,on ,,slide ,,rules--,,tables ,,& ,,data--,,slide ,,rule ,,data ,,slips4 ,,! ,,pickwor? ,,slide ,,rule4--,9 ?is rule1 made by ,mr4 ,a4 ,w4 ,fab]1 ! novel feature is ! provision ( a scale ( cubes 7,f7 9 ! /ock or body ( ! rule4 ,f ,fig4 #cc x w be se5 t ! scale is fix$ on ! bevell$ side ( a slott$ recess 9 ! back ( ! rule4 ,! slide c>ries an index m>k1 :i* is se5 ?r\gh ! slot & c be set to any graduation ( ! scale2 9 its normal position x agrees ) #a on ! scale4 ,! ,c scale on ! face ( ! rule is divid$ into ?ree equal p>ts by two special division l9es1 m>k$ ,,ii4 & ,,iii41 :i*1 toge!r ) ! initial graduation #a ( ! scale1 s]ve = sett+ or read+ (f values on ! ,d scale4 ,simil> division l9es are m>k$ on ! ,d scale4 ,illu/ration3 ,,fig4 #cc4 ,9 us+ ! rule = cubes or cube roots ! slide is drawn to ! "r1 ?is movem5t nev] exce$+ "o-?ird ( ! l5g? ( ! ,d scale4 ,) ?is limit$ movem5t1 & ) a s+le sett+ ( ! slide1 ! values ( a1 7a #aj71 & 7a #ajj77 7a be+ less ?an #aj & n less ?an #a7 are giv5 simultane\sly & )\t any unc]ta9ty z to ! scales to use or ! values to be read (f4 ,to ,f9d ! ,cube ( a ,numb]4--,! m>ks ,,ii4 & ,,iii4 on ,d divide t scale into ?ree equal sections4 ,if ! numb] to be cub$ is 9 ! fir/ section1 ,i4 on ,c is set to x2 if 9 ! second section1 ,,ii4 on ,c is set to x2 if 9 ! ?ird section1 ,,iii4 on ,c is set to x4 ,!n1 "u ! index m>k on ! back ( ! slide w be f\nd ! significant figures ( ! cube on ! scale ,f4 ,if ,i4 on ,c 0 us$ = ! sett+1 ! cube conta9s #a digit2 if ,,ii4 0 us$1 #b digits2 if ,,iii4 0 us$1 #c digits4 ,if ! fir/ figure ( ! numb] to be cub$ is n 9 ! units place1 ! decimal po9t is mov$ ?r\gh n places s z to br+ ! fir/ significant figure into ! units place1 ! cube f\nd z above1 & ! decimal po9t mov$ 9 ! rev]se direction ?r\gh #cn places4 ,to ,f9d ! ,cube ,root ( a ,numb]4--,! index m>k is set to ! significant figures ( ! numb] on scale ,f1 & ! cube root is read on ,d "u ,i41 ,,ii4 or ,,iii4 on ,c1 accord+ z ! numb] has #a1 #b or #c digits prec$+ ! decimal po9t4 ,numb]s :i* h #a1 #b or #c figures prec$+ ! decimal po9t are dealt ) directly4 ,numb]s ( any o!r =m are br\ght to "o ( ! above =ms by mov+ ! decimal po9t #c places 7or su* multiple ( #c places z may be requir$71 ! root f\nd & its decimal po9t mov$ #a place = ea* #c-place movem5t1 b 9 ! rev]se direction4 ,,! 8,,electro0 ,,slide ,,rule4--,9 ?is special rule = electrical calculations1 made by ,mr4 ,a4 ,ne/l]1 ! upp] scales run f #ja to #ajjj1 & are m>k$ 8,amp40 & 8sq4 mm40 respectively4 ,! l[] scale on ! slide runn+ f #a to #aj1jjj is m>k$ ,m 7metres71 :ile ! l[] scale on ! rule 7#ja to #ajj7 is m>k$ 8,volt40 ,! latt] scale is s displac$ t #aj on ,m agrees ) #jagc on ! ,volt scale4 ,! f\r factors involv$ are ! curr5t /r5g? 79 ,amp472 ! area ( a conductor 79 sq4 mm472 ! l5g? ( ! conductor 79 metres72 & ! p]missible loss ( pot5tial 79 volts74 ,hav+ giv5 any ?ree ( ^!1 ! f\r? c be f\nd v readily4 ,on ! back ( ! slide are a scale ( squ>es1 a scale ( cubes & a s+le scale correspond+ to ! ,d scale ( an ord9>y rule4 ,h5ce1 by rev]s+ ! slide1 x is possible to obta9 ! #bnd1 #crd & #d? p[]s & roots ( numb]s4 ,9 ano!r =m ( ! rule1 ! scale ( metres is replac$ by "o ( y>ds1 :ile in/ead ( ! area ( ! conductor 9 sq4 mm41 ! correspond+ 8gauge0 sizes ( wires are giv54 ,,! 8,,polyphase0 ,,slide ,,rule4--,?is in/rum5t1 made by ! ,keuffel ,ess] ,company1 ,new ,york1 has1 9 addition to ! usual scales1 a scale ( cubes on ! v]tical edge ( ! /ock ( ! rule1 :ile 9 ! c5tre ( ! slide !re is a rev]s$ ,c scale2 i4e41 a scale exactly simil> to an ord9>y ,c scale b ) ! graduations runn+ f "r to left4 ,! rule is specially useful = ! solution ( problems conta9+ comb9ations ( ?ree factors & problems involv+ squ>es1 squ>e roots1 cubes1 cube roots & _m ( ! high] p[]s & roots4 ,x is specially adapt$ = electrical & hydraulic "w4 ,,! ,,log-,,log ,,duplex ,,slide ,,rule4--,! same mak]s h introduc$ a log-log duplex slide rule1 9 :i* ! log-log scale is 9 ?ree sections1 plac$ "o above ! o!r1 ^! occupy+ ! position usually tak5 up by ! ,a scale4 ,^! scales are us$ 9 ! mann] already describ$ 7page #hf71 b "s advantage is obta9$ by ! mann] 9 :i* ! complete log-log scale is divid$1 ! limits be+ e to e 7on ,scale ,l4,l4 #a72 e to e 7on ,scale ,l4,l4 #b72 & e to e#aj 7on ,scale ,l4,l4 #c71 e be+ ! base ( natural or hyp]bolic log>i?ms 7#bgahbh74 ,9 ?is way a total log-log range ( f #aja to #bb1jjj is provid$1 meet+ all practical requirem5ts4 ,^! log-log scales are read 9 conjunction ) a ,c scale plac$ at ! upp] edge ( ! slide4 ,a simil> ,c scale1 b rev]s$ 9 direction1 is plac$ at ! l[] edge ( ! slide1 ?is hav+ r$ figures to di/+ui% x readily4 ,! adjac5t scale on ! body ( ! rule is an ord9>y ,d scale1 & "u ?is is an equally-divid$ scale giv+ ! common log>i?ms ( values on ,d4 ,9 ! c5tre ( ! slide is a scale ( tang5ts4 ,x w be und]/ood t a 8duplex0 rule consi/s ( two side /rips securely clamp$ toge!r at ! two ends1 =m+ ! body ( ! rule1 ! slide mov+ betwe5 !m2 h5ce bo? front & back faces ( ! rule & slide are available1 graduations on ! "o side be+ ref]r$ to ^? on ! o!r by ! cursor :i* ext5ds ar\nd ! :ole4 ,9 ?is in/rum5t1 ! scales on ! back face are ! ord9>y scales ( ! /&>d rule ) ! addition ( a scale ( s9es :i* is plac$ 9 ! c5tre ( ! slide4 ,x w be evid5t t ?is in/rum5t is capable ( deal+ ) a v wide range ( problems involv+ expon5tial & trigonometrical =mul4 ,,small ,,slide ,,rules ,,) ,,magnify+ ,,cursors4--,sev]al mak]s n[ supply #e 94 rules hav+ ! full graduations ( a #aj 94 rule1 & fitt$ ) a magnify+ cursor 7,fig4 #cd74 ,?is =ms a compact in/rum5t = ! pocket1 b ow+ to ! clos5ess ( ! graduations x is n usually possible to make a sett+ ( ! slide )\t us+ ! cursor4 ,?is1 ( c\rse1 involves m movem5ts ?an ) ! ord9>y in/rum5t4 ,x is also v necess>y to use ! magnify+ cursor 9 a direct light1 if accurate read+s are to be obta9$4 ,if ^! slight inconv5i5ces are to be tol]at$1 ! pr9ciple c\ld be ext5d$1 a #aj 94 rule be+ m>k$ z fully z a #bj 941 & fitt$ ) a magnify+ cursor4 ,! au?or has endeav\r$1 b )\t success1 to induce mak]s to introduce su* a rule4 ,! magnify+ cursor1 suppli$ by ,messrs4 ,a4 ,g4 ,?ornton1 ,limit$1 has a l5s :i* fills ! entire cursor4 ,x has a p[]ful magnify+ effect1 & ! *ange f ! natural to ! magnifi$ read+ is less abrupt ?an ) ! semicircul> l5s4 ,illu/ration3 ,,fig4 #cd4 ,,! ,,*emi/',s ,,slide ,,rule4--,a slide rule1 specially adapt$ = *emical calculations1 has be5 introduc$ rec5tly by ,mr4 ,a4 ,ne/l]4 ,9 ?is in/rum5t ! ,c & ,d scales are z usually arrang$2 b1 9 place ( ! ,a & ,b scales1 !re are a numb] ( gauge po9ts or m>ks d5ot+ ! atomic & molecul> weights ( ! mo/ important elem5ts & comb9ations4 ,! scales on ! back ( ! slide are simil>ly arrang$1 s t by rev]s+ ! slide ! op]ations c be ext5d$ v consid]ably4 ,! rule f9ds its *ief use 9 ! calculation ( analyses4 ,?us1 to f9d ! p]c5tage ( *lor9e if s grammes ( a sub/ance h be5 us$ & ! precipitate ( ,ag4,cl4 weighs a grammes1 we h ! equation1 x 7,cl47_/7,ag4,cl47 7a7_/7s74 ,h5ce1 ! m>k ,ag4,cl4 on ! upp] scale ( ! slide is set to ! m>k ,cl4 on ! upp] scale ( ! rule1 :5 "u a on ! ,c scale is f\nd ! quantity ( *lor9e on ,d4 ,by sett+ ! cursor to ?is value & br++ s on ,c to ! cursor1 ! p]c5tage requir$ c be read on ,c ov] #aj on ,d4 ,! rule is also adapt$ to ! solution ( v>i\s o!r *emical & electro-*emical calculations4 ,,! ,,/elfox ,,slide ,,rule4--,?is rule1 %[n 9 ,fig4 #ce1 has a /ock #e 94 long1 fitt$ ) a #aj 94 slide jo9t$ 9 ! middle ( its l5g? by means ( long d[els4 ,by sep>at+ ! p>ts ! compactness ( a #e 94 rule is obta9$4 ,! upp] scales on ! rule & slide resemble ! usual ,a & ,b scales4 ,! ,d scale on ! l[] "p ( ! /ock is 9 two sections1 ! second portion be+ plac$ bel[ ! fir/1 z %[n 9 ! illu/ration4 ,! c5tre scale on ! slide corresponds to ! usual ,c scale1 :ile on ! l[] edge ( ! slide is a simil> scale1 b ) ! index 7#a7 9 ! middle ( its l5g?4 ,! arrangem5t avoids ! necessity ( resett+ ! slide1 z is sometimes necess>y ) ! ord9>y rule1 & 9 g5]al x comb9es ! accuracy ( a #aj 94 rule ) ! compactness ( a #e 94 rule2 b a m frequ5t use ( ! cursor is necess>y4 ,?is rule is made by ,messrs4 ,john ,davis ,son1 ,limit$1 ,d]by4 ,illu/ration3 ,,fig4 #ce4 ,,electrical ,,slide ,,rule4--,ano!r rule by ! same mak]s1 specially useful = electrical eng9e]s1 has ! usual scales on ! work+ edges ( ! rule & slide1 :ile 9 ! middle ( ! slide is plac$ a scale ( cubes4 ,a log-log scale 9 two sections is provid$2 ! p[] portion1 runn+ f #ajg to #b1 is f\nd on ! l[] "p ( ! /ock1 & ! upp] portion1 runn+ f #b to #ajc1 on ! upp] "p ( ! /ock4 ,! upp]mo/ scale on ! /ock is 9 two p>ts1 ( :i* t to ! left1 runn+ f #bj to #ajj & m>k$ 8,dynamo10 gives ! effici5cies ( dynamos2 t on ! "r1 runn+ f #bj to #ajj & m>k$ 8,motor10 gives ! effici5cies ( electric motors4 ,! l[e/ scale on ! /ock1 m>k$ 8,volt10 gives ! loss ( pot5tial 9 copp] conductors4 ,! ord9>y upp] scale on ! /ock is m>k$ ,l 7l5g? ( lead7 at ! left1 & ,,kw 7kil[atts7 at ! "r2 ! ord9>y upp] scale on ! slide is m>k$ ,a 7ampres7 & mm#b 7sectional area7 at ! left1 & ,,hp 7horse-p[]7 at ! "r4 ,additional l9es on ! cursor enable ! electrical calculations to be made ei!r 9 ,briti% or metric units4 ,,! ,,picolet ,,circul> ,,slide ,,rule4--,a simple =m ( circul> calculator1 made by ,mr4 ,l4 ,e4 ,picolet ( ,philadelphia1 is %[n 9 ,fig4 #cf4 ,x consi/s ( a base disc ( /\t celluloid on :i* turns a small] disc ( ?9 celluloid4 ,a cursor =m$ ( transp>5t celluloid is fold$ ov] ! discs1 & is atta*$ s t ! friction betwe5 ! cursor & ! inn] disc enables ! latt] to be turn$ by mov+ ! =m]4 ,by hold+ bo? discs ! cursor c be adju/$ z requir$4 ,! adjac5t scales run 9 opposite directions1 s t multiplication & division are p]=m$ z ) ! inv]t$ slide 9 an ord9>y rule4 ,! out] scale1 :i* is two-?irds ! l5g? ( ! ma9 scale1 enables cube roots to be f\nd4 ,squ>e roots are readily det]m9$ & cont9u\s multiplication & division conv5i5tly effect$4 ,modifi$ =ms ( ?is neatly made little in/rum5t are also available4 ,illu/ration3 ,,fig4 #cf4 ,,o!r ,,rec5t ,,slide ,,rules4--,among o!r special types ( slide rule1 m5tion %\ld be made ( ! ,jak9 #aj 94 rule = surveyors1 made by ,messrs4 ,john ,davis ,son1 ,limit$1 ,d]by4 ,by ! provision ( a s]ies ( %ort subsidi>y scales1 ! multiplication ( a s9e or tang5t ( an angle by a numb] c be obta9$ to an accuracy ( #a 9 #aj1jjj4 ,! ,davis-,lee-,bottomley slide rule1 by ! same mak]s1 has special scales provid$ = circle spac+4 ,! division ( a circle into a numb] ( equal p>ts1 (t5 requir$ 9 spac+ rivets1 bolts1 etc41 & 9 sett+ out ! tee? ( ge>:eels1 is readily effect$ by ! aid ( ?is in/rum5t4 ,! ,cuntz slide rule is a v compreh5sive in/rum5t1 hav+ a /ock ab\t #b 94 wide1 ) ! slide ne> ! l[] edge4 ,above ! slide are elev5 scales1 ref]able to ! ma9 scales by ! cursor4 ,^! scales enable squ>es & squ>e roots1 cubes & cube roots1 & areas & circumf]5ces ( circles to be obta9$ by direct read+4 ,a mu* m compact in/rum5t c\ld be obta9$ by remov+ "o-half ! scales to ! back ( ! rule & us+ a d\ble cursor4 ,illu/ration3 ,,fig4 #cg4 ,9 "o =m ( #aj 94 rule1 suppli$ by ,mr4 ,w4 ,h4 ,h>l+1 ,london1 ! body ( ! rule is made ( well-season$ cane1 ) ! usual celluloid fac+s4 ,! rule has a metal back1 enabl+ ! fit ( ! slide to be regulat$4 ,?is back+ ext5ds ! full l5g? ( ! rule1 op5+s ab\t #a 94 long be+ provid$ at ea* end1 enabl+ ! scales on ! back ( ! slide to be set ) great] facility ?an is possible ) ! not*$ recesses usually adopt$4 ,! au?or has long endeav\r$1 b )\t success1 to induce mak]s to fit w9d[s ( glass or celluloid 9 place ( ! not*$ recesses4 ,?is w\ld all[ ! graduation ( ! ,s & ,t scales to be set m accurately1 & enable bo? to be us$ at ea* end ( ! rule--an advantage 9 c]ta9 trigonometrical calculations4 ,x w\ld h ! fur!r advantage ( p]mitt+ ea* alt]nate graduation ( ! ev5ly-divid$ or log>i?m scale to be plac$ at opposite sides ( "o c5tral l9e1 enabl+ ! read+ to be made m accurately & conv5i5tly4 ,_m special slide rules h lately be5 devis$ = det]m9+ ! "t necess>y to p]=m v>i\s ma*9e-tool op]ations & = analog\s purposes1 :ile att5tion has aga9 be5 giv5 to rules = calculat+ ! weights ( iron & /eel b>s1 plates1 etc4 ,,! ,,davis-,,/okes ,,field ,,gunn]y ,,slide ,,rule4--,?is rule1 :i* is adapt$ = calculations involv$ 9 8enc\nt]0 & 8entr5*$0 field gunn]y1 is design$ = ! #ah pr4 quick-fir+ gun4 ,! upp] & l[] portions ( ! boxwood /ock are unit$ by a flexible c5tre ( celluloid1 ?us provid+ grooves front & re> to receive boxwood slides4 ,ea* ( ! n9ete5 scales is m>k$ ) its "n1 & correspond+ scales are col\r$ r$ or black4 ,! front edge is bevell$ & c>ries a scale ( #a 9 #bj1jjj4 ,! rule solves displacem5t problems1 map angles ( sight1 *anges ( corrector & range corrections = *anges 9 temp]ature1 w9d & b>omet]1 etc4 ,a special feature = displacem5t calculations is ! provision ( a #ej yd4 sub-base angle scale1 by :i* ! apex angle is read at "o sett+4 ,,! ,,davis-,,m>t9 ,,wireless ,,slide ,,rule4--,9 wireless telegraphy x is frequ5tly necess>y to det]m9e wave-l5g?1 capacity or self-induction :5 "o or o!r ( ! factors ( ! equation1 #eif7,,lc7 is unkn[n4 ,! ,davis-,m>t9 wireless rule is design$ to simplify su* calculations4 ,! upp] scale 9 ! /ock 7inductance7 runs f #aj1jjj to #a1jjj1jjj2 ! adjac5t scale on ! slide 7capacity7 runs f #jjjja to #jja b 9 ! rev]se direction4 ,! l[] scale on ! /ock 7wave-l5g?7 runs f #ajj to #ajjj1 giv+ squ>e roots ( ! upp] scale2 :ile on ! l[] edge ( ! scale are sev]al arr[s to suit ! v>i\s d5om9ations 9 :i* ! wave-l5g? & capacity may be express$4 ,,improv$ ,,cursors4--,9 "s slide-rule op]ations1 notably 9 ^? involv$ 9 solv+ quadratic & cubic equations1 x n infrequ5tly happ5s t read+s are obscur$ by ! frame ( ! cursor4 ,frameless cursors h be5 introduc$ to obviate ?is defect4 ,a piece ( ?ick transp>5t celluloid is sometimes employ$1 b ?is is liable to become scrat*$ 9 use4 ,fig4 #cg %[s a rec5t =m ( frameless glass cursor made by ! ,keuffel ,ess] ,company1 ,hobok51 ,n4,j41 :i* is satisfactory 9 e way4 ,cursors hav+ ?ree hair l9es are n[ fitt$ to "s rules1 ! di/ance ap>t ( ! l9es be+ equal to ! int]val #jghed-a on ! ,a scale4 ,,! ,,davis-,,pletts ,,slide ,,rule4--,9 ?is rule a s+le log4-log4 scale & its reciprocal scale are arrang$ opposite ! ord9>y upp] log4 scale4 ,?us1 common log>i?ms c be read directly1 :ile by tak+ advantage ( ! prop]ties ( *>act]i/ics & mantissas ( common log>i?ms1 ! scale c be ext5d$ indef9itely4 ,z #aj is ! highe/ numb] on ! log4-log4 scale1 x is c>ri$ d[n to )9 #jjbe ( unity4 ,! read+ ( log4-log4 values above #aj is effect$ 9 a v simple mann]4 ,!re is also a scale 9 ! c5tre ( ! slide :i*1 us$ 9 conjunction ) ! upp] log4 scale enables ! natural log>i?m ( any numb] betwe5 #jjjja & #aj1jjj to be read direct1 :ile any numb] on ! upp] log4 scale c be multipli$ or divid$ by ex if ! latt] is betwe5 ^! limits4 ,on ! back ( ! slide are scales = all circul> & hyp]bolic functions1 ^! be+ us$ 9 conjunction ) ! upp] log4 scales4 ,,! ,,crompton-,,gallagh] ,,boil] ,,effici5cy ,,calculator has a /ock 9 ! ?ickness ( :i* is a slot admitt+ a *>t :i* c be mov$ at "r angles to ! two sep>ate slides4 ,on ! bevell$ edge ( "o slide1 ! graduations are cont9u$ s z to read aga9/ curves on ! *>t1 ?r\gh an op5+ 9 ! /ock4 ,,! ,,davis-,,gr9/$ ,,complex ,,calculator4--,?is slide rule is ( consid]able s]vice 9 connection ) calculations involv+ ! conv]sion ( complex quantities f ! =m a j b to ! =m ,r1 & vice v]sa4 ,! usual process ( conv]sion necessitates repeat$ ref]5ce to trigonometrical tables1 & is bo? t$i\s & "t-tak+4 ,! ,complex ,calculator enables ! conv]sions to be effect$ )\t ref]5ce to tables & ) ! m9imum exp5diture ( "t & lab\r4 ,! rule1 :i* is ab\t #af 94 long1 has five scales4 ,! upp] "o 7,a7 is an ord9>y log>i?mic scale ?rice-repeat$4 ,! adjac5t scales on ! slide comprise 7#a7 a log>i?mic scale ( tang5ts 7,b7 rang+ f #ja to #de1 & 7#b7 a log>i?mic scale ( secants 7,c7 f #j to #de4 ,! l[] scales ,d & ,e are id5tical ) ! ,a scale1 & are provid$ to enable multiplication1 etc41 to be p]=m$ )\t ! ne$ = a sep>ate slide rule4 ,read+s c be transf]r$ f ,a to ! l[] scales by means ( ! cursor4 ,9 us+ ! rule to conv]t a j b to ,r1 ! index 7#de7 ( ! ,b scale is set to ! l>g] compon5t & ! cursor to ! small] compon5t1 on scale ,a4 ,!n 7or its complem5t if b is great] ?an a7 is read on ,b "u ! cursor4 ,! cursor is !n set to on ! ,c scale1 & ,r is read on ,a "u ! cursor4 ,! rule is made by ,messrs4 ,john ,davis ,son1 ,limit$1 ,d]by4 ,,! ,,solution ,,( ,,algebraic ,,equations4 ,! slide rule f9ds an int]e/+ application 9 ! solution ( equations ( ! second & ?ird degree2 & al?\gh ! process is ess5tially "o ( trial & error1 x may (t5 s]ve z an effici5t sub/itute = ! m labori\s algebraic me?ods1 p>ticul>ly :5 ! conditions ( ! problem or ! op]ator's k ( ! !ory ( equations enables "s idea to be obta9$ z to ! "* ( ! result s\ght4 ,! pr9ciple may be ?us briefly expla9$3--,if #a on ,c is set to x on ,d 7,fig4 #ch71 we f9d x7x7 x#b on ,d "u x on ,c4 ,if1 h[ev]1 ) ! slide set z be=e1 in/ead ( read+ "u x1 we read "u x m on ,c1 ! result on ,d w n[ be x7x m7 x#b mx q4 ,h5ce to solve ! equation x#b mx q #j1 we rev]se ! above process1 & sett+ ! cursor to q on ,d1 we move ! slide until ! numb] on ,c "u ! cursor1 & t on ,d "u #a on ,c1 diff] by m4 ,x is obvi\s f ! sett+ t ! product ( ^! numb]s q1 & z _! diff]5ce m1 !y are se5 to be ! roots ( ! equation z requir$4 ,= ! equation x#b mx q #j1 we require m to equal ! sum ( ! roots4 ,h5ce1 sett+ ! cursor z be=e to q on ,d1 we move ! slide until ! numb] on ,c "u ! cursor1 & t on ,d "u #a on ,c1 are toge!r equal to m1 ^! numb]s be+ ! roots s\ght4 ,! alt]native equations x#b mx q #j1 & x#b mx q #j are d$ucible f ! o!rs by *ang+ ! signs ( ! roots1 & ne$ n be fur!r consid]$4 ,illu/ration3 ,,fig4 #ch4 ,,ex4--,f9d ! roots ( x#b #hx #i #j4 ,set ! cursor to #i on ,d1 & move ! slide to ! "r until :5 #ffd is f\nd "u ! cursor1 #acee on ,d is "u #a on ,c4 ,^! numb]s are ! roots requir$4 ,! upp] scales c ( c\rse be us$2 inde$1 9 g5]al !y are to be pref]r$4 ,,ex4--,f9d ! roots ( x#b #abhx #cid #j4 ,set ! cursor to #cid on ,a1 & move ! slide to ! "r until we read #gfe on ,b "u ! cursor1 & #eae on ,a ov] #a on ,b4 ,! roots are !re=e #gfe & #e4ae4 ,) a little consid]ation ( ! relative value ( ! upp] & l[] scales1 ! /ud5t int]e/$ w readily p]ceive h[ equations ( ! ?ird degree may be simil>ly resolv$4 ,! subject is n ( suffici5t g5]al importance to w>rant a detail$ exam9ation be+ made ( ! sev]al expressions :i* c be dealt ) 9 ! mann] sugge/$2 b ! au?or gives ! foll[+ example z af=d+ "s indication ( ! adaptability ( ! me?od to practical calculations4 ,,ex4--,a holl[ copp] ball1 #ge 94 9 diamet] & #b lb4 9 weight1 floats 9 wat]4 ,to :at dep? w x s9k8 ,! wat] displac$ #bgg #b #eed cub4 94 ,! cubic cont5ts ( ! imm]s$ segm5t w be 77_/7#c77cr x#b x#c71 r be+ ! radius & x ! dep? ( imm]sion4 ,h5ce 77_/7#c77cr x#b x#c7 #eed1 & #aabex#b x#c #ebi4 ,to solve ?is equation we place ! cursor to #ebi on ,a1 & move ! slide until ! read+ on ,d "u #a & t on ,b "u ! cursor toge!r am\nt to #aabe4 ,9 ?is way f9d #bde on ,d "u #a1 ) #hh on ,b "u ! cursor c1 c1 z a pair ( values ( :i* ! sum is #aabe4 ,h5ce we conclude t x #bde 94 is ! result s\ght4 ,) ! rule ?us set 7,fig4 #ci7 ! /ud5t w note t ! slide is displac$ to ! "r by an am\nt :i* repres5ts x on ,d1 & !re=e x#b on ,a2 :ile ! l5g? on ,b f #a to ! cursor l9e repres5ts #aabe x4 ,h5ce ! upp] scale sett+ gives x#b7aabe x7 #aabex#b x#c #ebi z requir$4 ,illu/ration3 ,,fig4 #ci4 ,:5 9 d\bt z to ! me?od to be pursu$ 9 any giv5 case1 ! /ud5t %\ld "w syn!tically1 build+ up a simple example ( an analog\s "* to t "u consid]ation1 & s d$uc+ ! plan to be foll[$ 9 ! rev]se process4 ,,screw-,,cutt+ ,,ge> ,,calculations4 ,! slide rule has long f\nd a useful application 9 connection ) ! ge> calculations necess>y 9 screw-cutt+1 helical ge>-cutt+1 & spiral ge> "w4 ,,s+le ,,ge>s4--,= simple cases ( screw-cutt+ 9 ! la! x is only necess>y to set ! ?reads p] in* to be cut to ! ?reads p] in* 9 ! guide screw 7or ! pit* 9 in*es 9 ea* case1 if m conv5i5t74 ,!n any pair ( co9cid+ values on ! two scales w give possible pairs ( :eels4 ,,ex4--,f9d :eels to cut a screw ( #a ?reads p] in* ) a guide screw ( #b ?reads p] in*4 ,sett+ #afbe on ,c to #b on ,d1 x is se5 t #hj 7driv]7 & #fe 7driv57 are possible :eels4 ,,comp\nd ,,ge>s4--,:5 :eels s f\nd are ( inconv5i5t size1 a comp\nd tra9 is us$1 consi/+ 7usually7 ( two driv]s & two driv5 :eels1 ! product ( ! two =m] & ! product ( ! two latt] be+ 9 ! same ratio z ! simple :eels4 ,?us ) #fj & #dj z driv]s1 & #fe & #cj z driv51 we h1 7#fj #dj7_/7fe #cj7 7#bdjj7_/7aiej7 7#b7_/7afbe7 z be=e4 ,) ! slide set z above1 values conv5i5t = splitt+ up into suitable :eels are readily obta9able4 ,?us1 7#afjj7_/7acjj72 7#bdjj7_/7aiej72 7#djjj7_/7cbej72 7#dhjj7_/7cijj7 are a few sugge/ive values :i* may be readily factoris$4 ,,slide ,,rules ,,= ,,screw-,,cutt+ ,,calculations4--,special circul> & /raight slide rules = screw-cutt+ ge> calculations h long be5 employ$4 ,= comp\nd ge>s ^! usually entail ! use ( six scales1 two on ea* ( ! two slides & two on ! /ock4 ,! upp] scale on ! /ock may be a scale ( ?reads p] in* to be cut1 ! adjac5t scale 7on ! upp] slide7 a scale ( ?reads p] in* 9 ! guide screw4 ,sett+ ! guide screw-graduation to ! ?reads to be cut1 ! l[] slide is adju/$ until a conv5i5t pair ( driv]s is f\nd 9 co9cid5ce on ! c5tral pair ( scales1 :ile a pair ( driv5 :eels are 9 co9cid5ce on ! two l[] scales4 ,"s ye>s ago1 a slide rule 0 introduc$ by :i* comp\nd ge>s c\ld be obta9$ ) a s+le slide4 ,assum+ ! set ( :eels usually provid$--#bj to #abj tee? advanc+ by #e tee?--! products ( #bj #be1 #bj #cj1 etc41 up to #aae #abj 7 calculat$4 ,^! products 7 laid out along ea* ( ! two l[] scales4 ,! upp] scales 7 a scale ( ?reads p] in* to be cut & a scale ( ! ?reads p] in* ( v>i\s guide screws4 ,sett+ ! guide screw-graduation to ! ?reads to be cut1 any co9cid+ graduations on ! l[] scales gave ! requir$ pairs ( driv]s & driv5 :eels4 ,,fractional ,,pit* ,,calculations4--,! au?or has long advocat$ ! use ( ! slide rule = det]m9+ ! :eels necess>y = cutt+ fractional pit* ?reads1 & x is gratify+ to f9d its value 9 ?is connection is n[ be+ appreciat$4 ,= ! be/ results a good #bj 94 rule is desirable1 b ) c>e v close approximations c be f\nd ) an accurate #aj 94 rule4 ,9 any case a magnify+ cursor or a h& read+-glass is ( great assi/ance4 ,,ex4--,f9d :eels to cut a ?read ( #jgjiji 94 pit*2 guide screw1 #b ?reads p] in*4 ,to #jgjiji on ,d1 set #je 7guide screw pit* 9 in*es7 on ,c4 ,to make ?is sett+ z accurately z possible1 ! me?od describ$ on page #aab may be us$4 ,set #aj on ,c to ab\t #ia on ,d1 & note t ! int]val #gg-gh on ,c repres5ts #jia ( ! int]val #gj-ga on ,d4 ,set ! cursor to #gh on ,c & br+ #e to ! cursor4 ,! slide is !n set s t #e on ,c agrees ) #gjia on ,d4 ,inspection ( ! two scales %[s v>i\s co9cid+ factors 9 ! ratio requir$4 ,! mo/ accurate is se5 to be 7#ee on ,c7_/7#gh on ,d74 ,^! values may be split up into 7#ee #ej7_/7fe #fj7 to =m a suitable comp\nd tra9 ( ge>s4 ,,gauge ,,po9ts ,,& ,,signs ,,on ,,slide ,,rules4 ,_m slide rules h ! sign 7,prod47_/7#a7 at ! "r-h& end ( ! ,d scale1 :ile on ! left is 7,quot47_/7#a47 ,x is some:at un=tunate t ^! signs ref] to rules = det]m9+ ! numb] ( digits 9 products & quoti5ts1 :i* are us$ to a consid]able ext5t on ! ,cont95t1 & conflict ) ^? us$ 9 ?is c\ntry4 ,by ! ,cont95tal me?od ! numb] ( digits 9 a product is equal to ! sum ( ! digits 9 ! two factors1 if ! result is obta9$ on ! ,,left ( ! fir/ factor2 b if ! result is f\nd on ! ,,"r ( ! fir/ factor1 x is equal to ?is sum #a4 ,! sign 7,prod47_/7#a7 ! "r-h& end ( ! ,d scale provides a visible rem9d] ( ?is rule4 ,simil>ly = division3--,! numb] ( digits 9 a quoti5t is equal to ! numb] ( ! digits 9 ! divid5d1 m9us ^? 9 ! divisor1 if ! quoti5t appe>s on ! ,,"r ( ! divid5d1 & to ?is diff]5ce #a1 if ! quoti5t appe>s on ! ,,left ( ! divid5d4 ,! sign 7,quot47_/7#a7 at ! left-h& end ( ! ,d scale provides a visible rem9d] ( ?is rule4 ,! sign - - f\nd at bo? ends ( ! ,a scale is ( g5]al application b ( que/ionable utility4 ,x is assum$ to repres5t a fraction1 ! v]tical l9e indicat+ ! position ( ! decimal po9t4 ,if ! numb] #dee is to be dealt ) 9 a multiplication on ! l[] scales1 we may suppose ! decimal po9t mov$ two places to ! left1 giv+ #dee1 a value :i* c be actually f\nd on ! scale4 ,if we use ?is value1 !n to ! numb] ( digits 9 ?is result1 z _m mu/ be add$ z ! numb] ( places 7two 9 ?is case7 by :i* ! decimal po9t 0 mov$4 ,if ! po9t is mov$ to ! "r1 ! numb] ( places mu/ be subtract$4 ,simil>ly1 9 division1 if ! decimal po9t 9 ! divisor is mov$ n places to ! left1 !n n places mu/ be subtract$ at ! end ( ! op]ation2 :ile if ! po9t is mov$ ?r\gh n places to ! "r1 !n n places mu/ be add$4 ,! sign ref]r$ to1 :i*1 ( c\rse1 applies to all scales1 completely indicates ^! processes & is submitt$ z a rem9d] ( ! proc$ure to be foll[$ by ^? us+ ! me?od describ$4 ,! signs 1 c1 c1 & ,m are expla9$ 9 ! ,section on 8,gauge ,po9ts10 p4 #ec4 ,on "s rules additional signs are f\nd on ! ,d scale4 ,"o1 locat+ ! value 7#ahj #fj7_/77 #cdcggd & h5ce giv+ ! numb] ( m9utes 9 a radian1 is m>k$ 4 ,ano!r1 repres5t+ ! value 7#ahj #fj #fj7_/77 #bjfbfe1 & h5ce giv+ ! numb] ( seconds 9 a radian is m>k$ 4 ,a ?ird po9t1 m>k$ 1 plac$ at ! value 7#bjj #ajj #ajj7_/77 #fcffbj1 is us$ :5 ! new] graduation ( ! circle is employ$4 ,^! gauge po9ts are useful :5 conv]t+ angles into circul> measure1 or vice v]sa1 & also = det]m9+ ! functions ( small angles4 ,a gauge po9t is sometimes m>k$ at #aadf on ! ,a & ,b scales4 ,?is is kn[n z ! 8,gunn]'s ,m>k10 & is us$ 9 artill]y calculations involv+ angles ( less ?an #bj1 :51 = ! purpose 9 view1 ! tang5t & circul> measure ( ! angle may be reg>d$ z equal4 ,= ?is con/ant1 ! angle is tak5 9 m9utes1 ! auxili>y base 9 feet1 & ! base 9 y>ds4 ,! auxili>y base 9 feet on ,b is set to ! angle 9 m9utes on ,a :5 ov] #aadf on ,b is ! base 9 y>ds on ,a4 ,! value 7#a7_/7aadf7 7 #c7_/7ahj #fj74 ,,tables ,,& ,,data4 ,,m5suration ,,=mulae4 ,area ( a p>allelogram base height4 ,area ( rhombus product ( ! diagonals4 ,area ( a triangle base p]p5dicul> height4 ,area ( equilat]al triangle squ>e ( side #jdcc4 ,area ( trapezium sum ( two p>allel sides height4 ,area ( any "r-l9$ figure ( f\r or m unequal sides is f\nd by divid+ x into triangles1 f9d+ area ( ea* & add+ toge!r4 ,area ( regul> polygon 7#a7 l5g? ( "o side numb] ( sides radius ( inscrib$ circle2 or 7#b7 ! sum ( ! triangul> areas into :i* ! figures may be divid$4 ,circumf]5ce ( a circle diamet] #cadaf4 ,circumf]5ce ( circle circumscrib+ a squ>e side #dddc4 ,circumf]5ce ( circle side ( equal squ>e #cede4 ,l5g? ( arc ( circle radius degrees 9 arc #jjagde4 ,area ( a circle squ>e ( diamet] #jghed4 ,area ( sector ( a circle l5g? ( arc radius4 ,area ( segm5t ( a circle area ( sector area ( triangle4 ,side ( squ>e ( area equal to a circle diamet] #jhhfb4 ,diamet] ( circle equal 9 area to squ>e side ( squ>e #aabhd4 ,side ( squ>e inscrib$ 9 circle diamet] ( circle #jgjg4 ,diamet] ( circle circumscrib+ a squ>e side ( squ>e #adad4 ,area ( squ>e area ( inscrib$ circle #abgcb4 ,area ( circle circumscrib+ squ>e squ>e ( side #aegjh4 ,area ( squ>e area ( circumscrib+ circle #jfcff4 ,area ( a p>abola base x height4 ,area ( an ellipse major axis m9or axis #jghed4 ,surface ( prism or cyl9d] 7area ( two ends7 7l5g? p]imet]74 ,volume ( prism or cyl9d] area ( base height4 ,surface ( pyramid or cone 7slant height p]imet] ( base7 area ( base4 ,volume ( pyramid or cone 777area ( base p]p5dicul> height74 ,surface ( sph]e squ>e ( diamet] #cadaf4 ,volume ( sph]e cube ( diamet] #jebcf4 ,volume ( hexagonal prism squ>e ( side #beih height4 ,volume ( p>aboloid volume ( circumscrib+ cyl9d]4 ,volume ( r+ 7circul> section7 mean diamet] ( r+ #bdg squ>e ( diamet] ( section4 ,,specific ,,gravity ,,& ,,weight ,,( ,,mat]ials4 ,,metals4 ,,metal4 ,specific ,weight ( #a ,weight ( #a ,gravity4 ,cub4 ,ft4 7,lb474,cub4 ,94 7,lb474 ,alum9ium1 ,ca/ #bef #afj #jjibg ,alum9ium1 ,bronze #gfh #dge #jbge ,antimony #fga #dah #jbdb ,bismu? #iij #fag #jceg ,brass1 ,ca/ #haj #eje #jbic ,wire #hedh #ecc #jcji ,copp]1 ,%eet #hhje #edi #jcah ,wire #hhhj #eed #jcba ,gold #aibde #abjj #jfie ,gun metal #hef #ecd #jcaj ,iron1 ,wr\ght 7mean7 #gfih #dhj #jbgh ,ca/ 7mean7 #gbag #dej #jbfa ,lead1 ,mill$ ,%eet #aadah #gab #jdab ,manganese #hjab #dii #jbhi ,m]cury #aceif #hdi #jdia ,nickel1 ,ca/ #hbh #eaf #jcjj ,phosphor ,bronze1 ,ca/ #hfj #ecfh #jcaj ,plat9um #baebb #acdb #jggh ,silv] #ajeje #fee #jchj ,/eel 7mean7 #gheb #dhif #jbhc ,t9 #gdji #dfb #jbfh ,z9c1 ,%eet #gbj #ddi #jbfj ,ca/ #fhf #dbh #jbdh ,,miscellane\s ,,sub/ances4 ,,sub/ance4 ,specific ,weight ( ,gravity4 #a ,cub4 ,94 7,lb474 ,asbe/os #ba-bhj #jgf-aja ,brick #aij #jfi ,cem5t #bgb-cje #jihd-aji ,clay #bj #jgb ,coal #acg #jdie ,coke #je #jaha ,concrete #bj #jgb ,fire-brick #bcj #jhc ,granite #be-bge #jea-ajj ,graphite #ah-bce #jfe-jhe ,s&-/one #bc #jhc ,slate #bh #ajb ,wood-- ,bee* #jge #jbga ,cork #jbd #jjhg ,elm #jeh #jba ,fir #jef #jbjc ,oak #fb-he #jbe-jca ,p9e #jdg #jag ,teak #jhj #jbi ,,ultimate ,,/r5g? ,,oe ,,mat]ials4 ,,mat]ial4 ,t5sion 9 ,compression ,%e>+ 9 ,modulus ( lb4 p] sq4 9 lb4 p] lb4 p] sq4 ,ela/icity 94 sq4 94 94 9 lb4 p] sq4 94 ,ca/ ,iron #aa1jjj to #ej1jjj to #ad1jjj1jjj #cj1jjj #acj1jjj to #bc1jjj1jjj av]4 #af1jjj #ie1jjj #aa1jjj ,wr\ght ,iron #dj1jjj to #bf1jjj1jjj #gj1jjj to #ca1jjj1jjj av]4 #ej1jjj #ej1jjj #dj1jjj ,s(t ,/eel #fj1jjj to #cj1jjj1jjj #ajj1jjj to #cf1jjj1jjj ,s(t ,/eel av]4 #hj1jjj #gj1jjj #ee1jjj ,ca/ ,/eel av]4 #abj1jjj #ae1jjj1jjj to #ag1jjj1jjj ,copp]1 ,ca/ #ai1jjj #eh1jjj ,wr\ght #cd1jjj #af1jjj1jjj ,brass1 ,ca/ #ah1jjj #aj1ejj #i1agj1jjj ,gun ,metal #cd1jjj #aa1ejj1jjj ,phosphor ,bronze #eh1jjj #dc1jjj #ac1ejj1jjj ,wood1 ,a% #ag1jjj #i1cjj #a1djj ,bee* #af1jjj #h1ejj ,p9e #aa1jjj #f1jjj #fej #a1djj1jjj ,oak #ae1jjj #aj1jjj #b1cjj #a1ejj1jjj ,lea!r #d1bjj #be1jjj ,,p[]s1 ,,roots1 ,,etc41 ,,( ,,useful ,,factors4 n 7#a7_/7n7 n#b n#c n 7#a7_/7n7 n 7#a7_/7n7 #cadb #jcah #ihgj #cajjf #aggb #jefd #adfe #jfhc #b #fbhc #jaei #cidgh #bdhjej #bejg #jcii #ahde #jedb 77_/7#b7 #aega #jfcg #bdfg #chgh #abec #jgih #aafb #jhfj 77_/7#c7 #ajdg #jiee #ajig #aadh #ajbc #jigg #ajaf #jihe 7#d7_/7c7 #dahi #jbci #agedf #gcdif #bjdg #jdhi #afab #jfbb 77_/7#d7 #jghe #abgd #jfag #jdhd #jhhf #aabh #jibc #ajhd 77_/7#f7 #jebd #aiaj #jbgd #jadd #jgbd #achb #jhjf #abda #b #ihgj #jaja #igdji #ifacij #cadb #jcah #bade #jdff #c #cajjf #jjcbifacijbi1hjiiaj #eefh #agif #cadb #jcah 77_/7#cb7 #jjih #ajahf #jjjie #jjja #jcac #caib #jdfa #bafh g #cbb #jjcaajcfhd #cc1chfbd #efgd #jagf #caha #jcad #b;g #fdd #jjaedadgcf #bfg1jij #hjbe #jabe #djjg #jbdi ,,hydraulic ,,equival5ts4 #a foot head #jdcd lb4 p] squ>e in*4 #a lb4 p] squ>e in* #bca ft4 head4 #a imp]ial gallon #bggbgd cubic in*es4 #a imp]ial gallon #jafjde cubic foot4 #a imp]ial gallon #aj lb4 #a cubic foot ( wat] #fbcb lb4 #fbcb imp]ial gallons4 #a cubic foot ( sea wat] #fdjj lb4 #a cubic in* ( wat] #jjcfaf lb4 #a cubic in* ( sea wat] #jjcgjcg lb4 #a cyl9drical foot ( wat] #dhif lb4 #a cyl9drical in* ( wat] #jjbhd lb4 ,a column ( wat] #ab 94 long #a 94 squ>e #jdcd lb4 ,a column ( wat] #ab 94 long #a 94 diamet] #jcdj lb4 ,capacity ( a #ab 94 cube #fbcb gallons4 ,capacity ( a #a 94 squ>e #a ft4 long #jjdcd gallon4 ,capacity ( a #a ft4 diamet] #a ft4 long #dhif gallons4 ,capacity ( a cyl9d] #a 94 diamet] #a ft4 long #jjcd gallon4 ,capacity ( a cyl9drical in* #jjjbhcb gallon4 ,capacity ( a cubic in* #jjjcfjf gallon4 ,capacity ( a sph]e #ab 94 diamet] #cbfc gallons4 ,capacity ( a sph]e #a 94 diamet] #jjjahh gallon4 #a imp]ial gallon #ab ,unit$ ,/ates gallon4 #a imp]ial gallon #dedc litres ( wat]4 #a ,unit$ ,/ates gallon #bcaj cubic in*es4 #a ,unit$ ,/ates gallon #jhc imp]ial gallon4 #a ,unit$ ,/ates gallon #ch litres ( wat]4 #a cubic foot ( wat] #gdgf ,unit$ ,/ates gallons4 #a cubic foot ( wat] #bhcge litres ( wat]4 #a litre ( wat] #jbb imp]ial gallon4 #a litre ( wat] #jbfd ,unit$ ,/ates gallon4 #a litre ( wat] #faj cubic in*es4 #a litre ( wat] #jjcec cubic foot4 ,,equival5ts ,,( ,,p\nds ,,avoirdupois4 #aj #ajj #ajjj #aj1jjj #ajj1jjj qr4 lb4cwt4 qr4 lb4ton cwt4 qr4 lb4ton cwt4 qr4 lb4ton cwt4 qr4 lb4 #a #j #aj #j #c #af #j #h #c #bj #d #i #a #d #dd #ab #c #ab #b #j #bj #a #c #d #j #ag #c #ab #h #ah #b #h #hi #e #b #bd #c #a #b #b #b #bj #a #f #c #d #ac #g #c #abacc #ah #b #h #d #a #ab #c #b #h #a #ae #b #bd #ag #ag #j #afagh #aa #a #bj #e #a #bb #d #a #bd #b #d #b #af #bb #f #a #bjbbc #d #a #d #f #b #d #e #a #ab #b #ac #b #h #bf #ae #b #bdbfg #ag #j #af #g #b #ad #f #a #j #c #b #b #j #ca #e #j #jcab #aj #j #j #h #b #bd #g #j #af #c #aa #a #bj #ce #ad #a #dceg #b #c #ab #i #c #f #h #j #d #d #j #a #ab #dj #c #b #hdja #ae #b #bd ,,trigonometrical ,,functions4 ,,"r-,,angl$ ,,triangles4 ,illu/ration3 ,"r-angl$ ,triangle ,s94 ,a 7a7_/7b7 ,sec4 ,a 7b7_/7c7 ,tan4 ,a 7a7_/7c7 ,cos4 ,a 7c7_/7b7 ,cosec4 ,a 7b7_/7a7 ,cotan4 ,a 7c7_/7a7 ,v]s94 ,a 7b c7_/7b74 ,cov]s94 ,a 7b a7_/7b74 ,giv54 ,requir$4 ,=mul4 a1b ,a1,c1c ,s94 ,a 7a7_/7b7 ,cos4 ,c 7a7_/7b7 c 77b a77b a77 a1c ,a1,c1b ,tan4 ,a 7a7_/7c7 ,cotan4 ,b 7a7_/7c7 b 7a#b c#b7 ,a1a ,c1c1b ,c #ij ,a c a ,cotan4 ,a b 7a7_/7,s94 ,a7 ,a1b ,c1a1c,c #ij ,a a b ,s94 ,a c b ,cos4 ,a ,a1c ,c1a1b ,c #ij ,a a c ,tan4 ,a b 7c7_/7,cos4 ,a7 ,,oblique-,,angl$ ,,triangles4 s 7a b c7 ,illu/ration3 ,oblique-angl$ ,triangle ,giv54 ,=mul4 ,a1,b1,c1a ,area 7a#b ,s94 ,b ,s94 ,c7 #b ,s94 ,a ,a1b1c 7c b ,s94 ,a7 a1b1c 7s7s a77s b77s c77 ,giv54 ,requir$4 ,=mul4 ,a1,c1a c c a7,s94 ,c7_/7,s94 ,a7 ,a1a1c ,c ,s94 ,c 7c ,s94 ,a7_/7a7 a1c1,b ,a ,tan4 ,a 7a ,s94 ,b7_/7c a ,cos4 ,b7 a1b1c ,a ,s94 ,a 777s b77s c77_/7b c77 ,cos4 ,a 77s7s a77_/7b c772 ,tan4 ,a 777s b77s c77_/7s7s a777 ,,comp\nd ,,angles4 ,s94 7,a ,b7 ,s94 ,a ,cos4 ,b ,cos4 ,a ,s94 ,b4 ,s94 7,a ,b7 ,s94 ,a ,cos4 ,b ,cos4 ,a ,s94 ,b4 ,cos4 7,a ,b7 ,cos4 ,a ,cos4 ,b ,s94 ,a ,s94 ,b4 ,cos4 7,a ,b7 ,cos4 ,a ,cos4 ,b ,s94 ,a ,s94 ,b4 ,tan4 7,a ,b7 7,tan4 ,a ,tan4 ,b7_/7#a ,tan4 ,a ,tan4 ,b74 ,tan4 7,a ,b7 7,tan4 ,a ,tan4 ,b7_/7#a ,tan4 ,a ,tan4 ,b74 ,,slide ,,rule ,,data ,,slips1 ,,compil$ ,,by ,c4 ,n4 ,,pickwor?1 ,,:4,,sc4 7,x is sugge/$ t ?is page be remov$ by cutt+ ?r\gh ! above l9e1 & select$ portions ( ! ,sectional ,data ,slips atta*$ to ! back ( ! ,slide ,rule47 #jjcabe #jjfbe #jjicge #jabe #jaefbe #jahge #jbahge #jbe #jbhabe #jcabe #jcdcge #jcge #jdjfbe #jdcge #jdfhge #jecabe #jefbe #jeicge #jfbe #jfefbe #jfhge #jgahge #jge #jghabe #jhabe #jhdcge #jhge #jijfbe #jicge #jifhge ,circ4 ( circle #cadaf d4 ,area #jghed d#b4 ,sq4 eq4 area to cir41 s #jhhf d4 ,circle eq4 to sq41 d #aabh s4 ,sq4 inscbd4 9 circ41 s #jgjg d4 ,circsb4 circ4 ( sq41 d #adad s4 ,area ( ellipse #j4ghed a b4 ,surface ( sph]e #cadaf d#b4 ,volume #jebcf d#c4 cone #jbfah d#b h4 ,radian 7#ahj7_/77 #egbi deg4 ,base ( nat4 or hyp4 log4 e #bgahc4 ,nat4 or hyp4 log4 com4 log4 #bcjbf4 g 7at ,london7 #cbah ft4 p] sec41 p] sec4 ,abs4 temp4 deg4 ,f4 #dfa deg4 ,c4 #bgd4 ,c4 7#e7_/7i77;,f4 #cb72 ,f4 7#i7_/7e7;,c4 #cb4 ,cal4 pr4--,!r4 units p] lb43 ,coal1 #ad1cjj2 petrol'm1 #bj1jjj2 coal gas p] cu4 ft41 #gjj4 ,sp4 heat3--,wt4 iron1 #jaach2 ,c4,i41 #jabih2 copp]1 brass1 #jjie2 lead1 #jjcad4 ,in* #bed mil'metres2 mil'metre #jjcicg 94 ,foot #jcjdh metres2 metre #cbhji feet4 ,y>d #jiadch metre2 metre #ajicf y>ds4 ,mile #afjic kilomtrs42 kilomtr4 #jfbac mile4 ,sq4 94 #fdeac sq4 cm42 sq4 cm4 #jaee sq4 94 ,sq4 ft4 #ibi sq4 decmtr42 sq4 decmtr4 #jajgf sq4 ft4 ,sq4 yd4 #jhcf sq4 metre2 sq4 metre #aaif sq4 yds4 ,sq4 ml4 #behi hect>es2 hect>e #jjjchf sq4 ml4 ,cu4 94 #afchf c4 cm42 c4 cm4 #jjfajb cu4 94 ,cu4 ft4 #jjbhc c4 metre2 c4 metre #cecaf cu4 ft4 ,gra9 #jjfdh gramme2 gram4 #aedc grs4 ,ounce #bhce grams42 #jjcebg oz4 ,p\nd #jdecf kilogm42 kilogm4 #bbjd lb4 ,ton #ajaf tonnes2 tonne #jihdb ton4 ,mile p] hr4 #adff ft41 or #ddg cm41 p] sec4 ,lb4 p] cu4 94 #jjbgf kilogram p] cu4 cm4 ,kilogram p] cu4 cm4 #cfabe lb4 p] cu4 94 ,lb4 p] cu4 ft4 #afjai kilogm4 p] cu4 mtre4 ,gra9 p] gall4 #jjadbf gramme p] litre4 ,gramme p] litre #gjaaf gra9s p] gall4 ,ultimate ,/r5g?,lb4 p] ,sq4 94 ,t5s'n4,comp'n4 ,wt4 iron #ej1jjj #ej1jjj ,ca/ #af1jjj #ie1jjj ,/eel #hj1jjj #gj1jjj ,copp] #ba1jjj #ej1jjj ,brass #ah1jjj #aj1ejj ,lead #b1ejj #g1jjj ,p9e #aa1jjj #f1jjj ,oak #ae1jjj #aj1jjj ,weight ( ,metals4 ,cub4 ,94 ,cub4 ,ft4 #ab ,cu4 ,94 ,wt4 iron #jbgg #dhj #ccc ,ca/ #jbfj #dej #cab ,/eel #jbhc #dij #cdj ,copp] #jcah #eej #chb ,brass #jcjj #ebj #cfa ,z9c #jbdh #dcj #bih ,alum9'm #j4jif #afh #aaf ,lead #j4daa #gaj #dic ,lb4 p] sq4 94 #bca ft4 wat] #bjd 94 m]cury #jjgjc kilo4 p] sq4 cm4 ,atmosph]e #adg lb4 p] sq4 94 #ccid ft4 wat] #ajcce ,ft4 hd4 wat] #jdcc lb4 p] sq4 94 #fbce lb4 p] sq4 ft4 #jjcjd ,cub4 ft4 ( wat] #fbce lb4 #jjbgh ton #bhcae litres #gdh ,u4,s4 galls4 ,gall4 7,imp47 #bggbg cu4 94 #jafjd cu4 ft4 #aj lb4 wat] #dedd litres4 ,litre #agf p9ts #jbb gall4 #fa cu4 94 #jjcec cu4 ft4 #jbfd ,u4,s4 gall4 ,horse-p[] #cc1jjj ft4-lb4 p] m94 #jgdf kil[att #dbd heat units p] m94 ,heat unit #ggh ft4-lb4 #ajee watt-sec4 #ajge kilogrammetres #jbeb calorie4 ,foot-p\nd #jjjabi heat unit #acf j\les #jachc kilogrammetres4 ,kil[att #acd ,h4,p4 #dd1bdj ft4-lb4 p] m94 #cdab heat units p] h\r4 ----- ,footnote #a3 ,x w be recognis$ t n is ! *>act]i/ic ( ! log>i?m ( ! orig9al numb]4 ,footnote #b3 ,! special case 9 :i* ! num]ator is #a1 #aj1 or any p[] ( #aj mu/ be treat$ by ! rule = reciprocals 7page #bg74 ,footnote #c3 ,! possible ne$ = trav]s+ ! slide1 to *ange ! indices1 :5 us+ ! ,c & ,d scales1 is n consid]$ z a sett+4 ,footnote #d3 ,! read] may be rem9d$ t cross-multiplication ( ! factors 9 any su* slide rule sett+ w give a con/ant product1 e4g41 #bj #ide #bg #gj4 ,footnote #e3 ,9 ?is case cross divid+ gives a con/ant quoti5t1 e4g41 #h #c #d #ae4 ,s9ce ! upp] scale is n[ a scale ( reciprocals1 ! ratio is really ,o ,d #ae #c ,footnote #f3 ,^! l9es %\ld n be br\ght to ! work+ edge ( ! scale b %\ld t]m9ate 9 ! horizontal l9e :i* =ms ! bord] ( ! f9] graduations1 _! value be+ read into ! calculation by means ( ! cursor 7see page #ee74 ,footnote #g3 ,! same pr9ciple may be appli$ to ! cursor4 ,footnote #h3 ,philosophical ,transactions ( ! ,royal ,society1 #ahae4 ---------------------------------------- -------------------------------- ,,by ,,! ,,same ,,au?or4 ,,log>i?ms ,,= ,,beg9n]s4 8,an extremely useful & mu*-ne$$ little "w1 giv+ a complete explanation ( ! !ory & use ( log>i?ms1 by a tea*] ( great cle>ness & good /yle40--,! ,m9+ ,j\rnal4 #as4 #h;d4 ,po/ ,free4 ,,! ,,indicator ,,h&book4 ,compris+ 8,! ,indicator3 ,its ,con/ruction & ,application0 & 8,! ,indicator ,diagram3 ,its ,analysis & ,calculation40 ,complete 9 ,"o ,volume4 #gs4 #aj;d4 ,po/ ,free4 8,mr4 ,pickwor?'s judgm5t is always s\nd1 & is evid5tly d]iv$ f a p]sonal acqua9tance ) indicator "w40--,! ,eng9e]4 ,,p[] ,,comput] ,,= ,,/eam1 ,,gas ,,& ,,oil ,,eng9es1 ,etc4 8,accurate1 exp$iti\s & ?or\ghly practical4444 ,we c confid5tly recomm5d x1 & eng9e]s w f9d x a great boon 9 und]tak+ te/s1 etc40--,! ,electrician4 #gs4 #f;d4 ,po/ ,free4 ---------------------------------------- -------------------------------- ,,adv]tisem5ts4 ,,log>i?ms ,,= ,,beg9n]s ,= a full & intellig5t appreciation ( ! ,slide ,rule & its v>i\s applications an elem5t>y k ( log>i?ms is necess>y4 ,all t is requir$ w be f\nd 9 ?is little "w1 :i* gives a simple1 detail$ & practical explanation ( log>i?ms & _! uses1 p>ticul> c>e hav+ be5 tak5 to elucidate all difficult po9ts by ! aid ( a numb] ( work$ examples4 ,sev5? ,edition1 #as4 #h;d4 ,po/ ,free4 ,,p[] ,,comput] = ,,/eam1 ,,gas1 ,,& ,,oil ,,eng9es1 ,etc4 ,gives ,! ,indicat$ ,horse-,p[] ( ,/eam1 ,gas1 & ,oil ,eng9es--,! ,brake ,horse-,p[] ( ,/eam1 ,gas1 & ,oil ,eng9es--,! ,size ( ,eng9e ,necess>y to ,develop any ,giv5 ,p[]--,! ,me*anical ,effici5cy ( an ,eng9e--,! ,ratio ( ,comp\nd ,eng9e ,cyl9d]s--,! ,pi/on ,spe$ ( an ,eng9e--,! ,deliv]y ( ,pumps ) any ,effici5cy--,! ,horse-,p[] ( ,belt+--,! ,rim ,spe$s ( ,:eels1 ,spe$s ( ,ropes1 ,belts1 etc4--,spe$ ,ratios ( ,pulleys1 ,ge>+1 etc4 ,pocket size1 9 neat case1 ) in/ructions & examples4 ,po/ ,free1 #gs4 #f;d4 net4 ,c4 ,n4 ,,pickwor?1 ,)+ton1 ,man*e/] ,w4 ,p4 ,,?ompson1 ,g4 ,c4 ,,dymond1 ,f4,c4,s41 ,m4,i4,me*4,e41 ,f4,i4,c4,p4,a4 ,m4,i4,me*4,e41 ,f4,i4,c4,p4,a4 ,w4 ,p4 ,?ompson ,co41 #ab ,,*ur* ,,/reet1 ,,liv]pool1 ,,*>t]$ ,,pat5t ,,ag5ts4 ,h4 ,e4 ,,potts1 ,j4 ,v4 ,,arm/rong1 ,m4,sc41 ,hon4 ,*em41 ,f4,i4,c4,p4,a4 ,m4,text4,i41 ,f4,i4,c4,p4,a4 ,w4 ,h4 ,,bee/on1 ,r4,p4,a4 ---------------------------------------- -------------------------------- ,,briti% ,,slide ,,rules = all ,,arts & ,,indu/ries includ+ ,illu/ration3 ,slide ,rule ,,log-,o-,,log ,,dr4 ,,yokota',s ,,surveyors' ,,wireless ,,gunn]y ,,electrical ,,rules1 ,etc4 ,,s5d ,,= ,,li/ #ee ,,made ,,by-- ,,john ,,davis ,,son 7,d]by71 ,ltd4 ,,all ,,sa9ts' ,,works1 ,,d]by ---------------------------------------- -------------------------------- ,k ,e ,slide ,rules are con/antly gr[+ 9 popul>ity1 & !y c n[ be obta9$ f ! lead+ h\ses 9 our l9e ?r\gh\t ! ,unit$ ,k+dom4 ,illu/ration3 ,slide ,rule ,we manufacture a complete l9e ( ,,eng9e-,,divid$ ,,slide ,,rules1 & call special att5tion to our ,pat5t ,adju/m5t1 ensur+ smoo? work+ ( ! ,slide2 also to our new 8,frameless0 ,indicator1 :i* hides no figures on ! ,rule4 ,illu/ration3 ,?a*]'s ,calculat+ ,in/rum5t ,,?a*]',s ,,calculat+ ,,in/rum5t1 = solv+ problems 9 multiplication1 division1 or comb9ations ( ! two2 has upw>ds ( #cc1jjj divisions4 ,results c be obta9$ to ! f\r? & usually to ! fif? place ( figures ) a surpris+ degree ( accuracy4 ,we also make ,,all ,,metal1 ,,circul>1 ,,/adia1 ,,*emi/s'1 ,,electrical1 & ,,o!r ,,special ,,slide ,,rules ,,descriptive ,,circul>s ,,on ,,reque/ ,,keuffel ,,ess] ,,co4 #abg ,fulton ,/41 ,,new ,,york ,g5]al ,(fice & ,factories1 ,,hobok51 ,n4,j4 ,,*icago ,,/4 ,,l\is ,,san ,,francisco ,,montreal ,,draw+ ,,mat]ials ,,ma!matical & ,,survey+ ,,in/rum5ts ,,measur+ ,,tapes ---------------------------------------- -------------------------------- ,illu/ration3 #f 94 ,/&>d ) magnify+ ,cursor complete 9 pocket case1 #e_/- ,,norton ,,gregory ,,ltd4 ,head ,(fice ,,ca/le ,,lane1 ,,we/m9/]1 ,,london1 ,s4,w4 #a ,bran*es #ga ,,que5 ,,/reet1 ,,glasg[4 ,,pho5ix ,,h\se1 ,,que5 ,,/reet & ,,s&hill1 ,,newca/le-,,on-,,tyne4 ,,slide ,,rules 9 ,/ock1 f #ag_/f to #bg_/f ,special ,quotations to ! ,trade = ,quantities ,= p>ticul>s ( ,survey+1 ,measur+ & ,ma!matical ,in/rum5ts1 ,appliances & ,mat]ial ( all k9ds = ! ,draw+ ,(fice1 write to ! ,head ,(fice4 ,illu/ration3 ,,norton ,,gregory ,,ltd ---------------------------------------- -------------------------------- ,,norton ,,gregory1 ,,ltd41 ,london4 8,,diamond0 ,,draw+ ,,in/rum5ts ,manufactur$ at our ,london ,works4 ,,c5tre ,,screw ,,spr+ ,,b[ ,,half ,,set4 ,illu/ration3 ,c5tre ,screw ,spr+ ,b[ ,half ,set #d in* ,spr+ ,b[ ,half ,set c5tre screw adju/m5t1 ) int]*angeable ne$le1 p51 & p5cil po9ts ,price #ag_/f ,! ,c5tre ,screw ,spr+ ,b[ ,half ,set ( ,compasses1 z illu/rat$1 possesses ! advantage ( ,,comb9+ ,,9 ,,"o ,,in/rum5t ,,! ,,set ,,( ,,?ree ,,sep>ate ,,spr+ ,,b[s hi!rto 9 use1 :ile ! c5tre screw makes = ease & accuracy ( manipulation1 at ! same "t provid+ a radius ( ov] #b in*es1 or d\ble t ( ! old patt]n4 ,?is in/rum5t is less exp5sive ?an ! set ( #c b[s1 :ile consid]ably /rong] 9 con/ruction4 ,! fix$ ne$le po9t is %\ld]$4 ,?is illu/ration is giv5 z an indication ( ! v>i\s ,draw+ ,in/rum5ts manufactur$ by u4 ,illu/rat$ ,booklet giv+ full p>ticul>s & prices ( o!r ,in/rum5ts & ,cases ( ,in/rum5ts s5t on application4 ,specially arrang$ ,sets ( ,in/rum5ts made = ,colleges1 ,s*ools1 ,te*nical ,in/itutes ,e/imates submitt$ on ,application4 ,write to our ,head ,(fice3 ,,ca/le ,,lane1 ,,we/m9/]1 ,,london1 ,s4,w4 #a4 ---------------------------------------- -------------------------------- ,,draw+ ,,& ,,survey+ ,,in/rum5ts ,a4 ,g4 ,,?ornton ,ltd4 ,,slide ,,rules ,,= ,p>agon ,works ,,accurate ,,sectional ,,eng9e]s #b ,k+ ,/4 ,we/ ,,pap]s ,,& ,,clo?s ,,man*e/] ,d #aiaf ,illu/rat$ ,catalogue1 j publi%$1 9 two editions2 ,draw+ ,(fice 7#ddh pages72 ,draughtsman's 7#afj pages73 ! mo/ complete ,catalogues 9 ! trade4 ,,contractors ,,to ,h4,m4 ,,w> ,,(fice ,,& ,,admiralty ,manufactur]s also ( ,draw+ ,mat]ials & ,draw+ ,(fice ,/ation>y4 7,,also ,,at ,,m9]va ,,works ,,& ,,alb]t ,,mills1 ,,man*e/]47 ,,ma!matical ,,in/rum5ts ,,survey+ ,,in/rum5ts ,,slide ,,rules ,= ,/ud5ts & ,eng9e]s ,,mannheim1 ,,polyphase1 ,,duplex1 ,,electrical1 ,,log-,,log1 ,,& ,,calculex ,j4 ,h4 ,,/ew>d ,,ltd4 ,sci5tific ,in/rum5t ,mak]s #djf ,,/r&1 & #deg ,,we/ ,,/r& ,,london1 ,w4,c4 #b ---------------------------------------- -------------------------------- ,a ,true ,fri5d & ,tru/y ,guide ,,! #8,,hald5 ,,calculex' ,illu/ration3 ,hald5 ,calculex ,,actual ,,size ,,briti% ,,made ,! h&ie/ & mo/ p]fect =m ( ,slide ,rule4 ,does all t c be done ) a /raight rule4 ,complete 9 ,case1 ) book ( in/ructions1 #bg_/f po/ free4 ,j4 ,,hald5 ,,co41 ,,ltd41 #h ,,alb]t ,,squ>e ,,man*e/] ,depots--,london1 ,newca/le-on-,tyne1 ,birm+ham1 ,glasg[1 & ,le$s ---------------------------------------- -------------------------------- ,illu/ration3 ,rope ,,eng9e]+1 ,,survey+ ,,& ,,ma!matical ,,in/rum5ts1 ,,etc4 ,,slide ,,rules4 ,,joseph ,,cas>telli ,,son1 #dc ,,m>ket ,,/reet1 ,,man*e/]4 ,tel4 ,no4 #bieh ,city4 ,e/abli%$ #agij4 ,,rope ,,driv+ ,is ! mo/ ,,effici5t & mo/ ,,economical ,,me?od ( ,p[] ,transmission4 ,! ,,lambe? ,cotton ,driv+ ,rope ,is ! mo/ ,,effici5t & mo/ ,,economical ,,rope = ,p[] ,transmission4 ,illu/ration3 ,made #d ,/r& or #c ,/r&4 ,,special ,,features3 ,,less ,,/ret* ,,?an ,,any ,,o!r ,,rope4 ,,m ,,pliable ,,?an ,,any ,,o!r ,,rope4 ,,great] ,,driv+ ,,p[] ,,?an ,,any ,,o!r ,,rope4 ,,?omas ,,h>t ,,ltd41 ,lambe? ,works1 ,,blackburn4 ---------------------------------------- -------------------------------- ,,transcrib]',s ,,notes ,page ,*ang$ f ,*ang$ to #bd ! "r1 s ! numb] ( ! "r1 s ! numb] ( digits 9 ! answ] #c #b #a digits 9 ! answ] #c #b #a #b #b #aaf grammes1 we h ! equation1 grammes1 we h ! equation1 x 7,cl47_/7,ag4,cl47 x 7,cl47_/7,ag4,cl47 7a7_/7s74 ,h5ce1 ! m>k 7a7_/7s74 ,h5ce1 ! m>k ,typos fix$2 non-/&>d spell+ & dialect reta9$4 ,us$ numb]s = footnotes1 plac+ !m all at ! end ( ! la/ *apt]4 ,enclos$ italics font 9 und]scores4 ,enclos$ bold font 9 equals4 ,! c>et 77 s]ves z a sup]script indicator1 applicable to individual *>act]s 7l #b;d7 & ev5 entire phrases 7l #a/74 ,subscripts are %[n us+ an und]score 77 ) curly braces 1 z 9 ,h#b,o4