第2章

类别:其他 作者:Katzenbach, John字数:20662更新时间:19/01/04 10:22:07
27。As,ontheonehand,itwereabsurdtogetridofobysaying,Letmecontradictmyself;letmesubvertmyownhypothesis;letmetakeitforgrantedthatthereisnoincrement,atthesametimethatIretainaquantitywhichIcouldneverhavegotatbutbyassuminganincrement:so,ontheotherhand,itwouldbeequallywrongtoimaginethatinageometricaldemonstrationwemaybeallowedtoadmitanyerror,thougheversosmall,orthatitispossible,inthenatureofthings,anaccurateconclusionshouldbederivedfrominaccurateprinciples。Thereforeocannotbethrownoutasaninfinitesimal,orupontheprinciplethatinfinitesimalsmaybesafelyneglected;butonlybecauseitisdestroyedbyanequalquantitywithanegativesign,whenceo-poisequaltonothing。Andasitisillegitimatetoreduceanequation,bysubductingfromonesideaquantitywhenitisnottobedestroyed,orwhenanequalquantityisnotsubductedfromtheothersideoftheequation:soitmustbeallowedaverylogicalandjustmethodofarguingtoconcludethatiffromequalseithernothingorequalquantitiesaresubductedtheyshallstillremainequal。Andthisisatruereasonwhynoerrorisatlastproducedbytherejectingofo。Whichthereforemustnotbeascribedtothedoctrineofdifferences,orinfinitesimals,orevanescentquantities,ormomentums,orfluxions。 28。Supposethecasetobegeneral,andthatisequaltotheareaABCwhencebythemethodoffluxionstheordinateisfound,whichweadmitfortrue,andshallinquirehowitisarrivedat。Nowifwearecontenttocomeattheconclusioninasummaryway,bysupposingthattheratioofthefluxionsofxandisfound[Sect。13。]tobe1and,andthattheordinateoftheareaisconsideredasitsfluxion,weshallnotsoclearlyseeourway,orperceivehowthetruthcomesout,thatmethodaswehaveshewedbeforebeingobscureandillogical。Butifwefairlydelineatetheareaanditsincrement,anddividethelatterintotwopartsBCFDandCFH,[Seethefigureinsect。26。]andproceedregularlybyequationsbetweenthealgebraicalandgeometricalquantities,thereasonofthethingwillplainlyappear。ForasisequaltotheareaABC,soistheincrementofequaltotheincrementofthearea,i。e。toBDHC;thatistosayAndonlythefirstmembersoneachsideoftheequationbeingretained,=BDFC:anddividingbothsidesbyoorBD,weshallget=BC。AdmittingthereforethatthecurvilinearspaceCFHisequaltotherejectaneousquantityandthatwhenthisisrejectedononeside,thatisrejectedontheother,thereasoningbecomesjustandtheconclusiontrue。AnditisallonewhatevermagnitudeyouallowtoBD,whetherthatofaninfinitesimaldifferenceorafiniteincrementeversogreat。Itisthereforeplainthatthesupposingtherejectaneousalgebraicalquantitytobeaninfinitelysmallorevanescentquantity,andthereforetobeneglected,musthaveproducedanerror,haditnotbeenforthecurvilinearspacesbeingequalthereto,andatthesametimesubductedfromtheotherpartorsideoftheequation,agreeablytotheaxiom,Iffromequalsyousubductequals,theremainderswillbeequal。Forthosequantitieswhichbytheanalystsaresaidtobeneglected,ormadetovanish,areinrealitysubducted。Ifthereforetheconclusionbetrue,itisabsolutelynecessarythatthefinitespaceCFHbeequaltotheremainderoftheincrementexpressedbyequal,Isay,tothefiniteremainderofafiniteincrement。 29。Therefore,bethepowerwhatyouplease,therewillariseononesideanalgebraicalexpression,ontheotherageometricalquantity,eachofwhichnaturallydividesitselfintothreemembers。Thealgebraicalorfluxionaryexpression,intoonewhichincludesneithertheexpressionoftheincrementoftheabscissanorofanypowerthereof;anotherwhichincludestheexpressionoftheincrementitself;andthethirdincludingtheexpressionofthepowersoftheincrement。Thegeometricalquantityalsoorwholeincreasedareaconsistsofthreepartsormembers,thefirstofwhichisthegivenarea;thesecondarectangleundertheordinateandtheincrementoftheabscissa;thethirdacurvilinearspace。And,comparingthehomologousorcorrespondentmembersonbothsides,wefindthatasthefirstmemberoftheexpressionistheexpressionofthegivenarea,sothesecondmemberoftheexpressionwillexpresstherectangleorsecondmemberofthegeometricalquantity,andthethird,containingthepowersoftheincrement,willexpressthecurvilinearspace,orthirdmemberofthegeometricalquantity。Thishintmayperhapsbefurtherextended,andappliedtogoodpurpose,bythosewhohaveleisureandcuriosityforsuchmatters。TheuseImakeofitistoshew,thattheanalysiscannotobtaininaugmentsordifferences,butitmustalsoobtaininfinitequantities,betheyeversogreat,aswasbeforeobserved。 30。Itseemstherefore,uponthewhole,thatwemaysafelypronouncetheconclusioncannotberight,ifinordertheretoanyquantitybemadetovanish,orbeneglected,exceptthateitheroneerrorisredressedbyanother;orthat,secondly,onthesamesideofanequationequalquantitiesaredestroyedbycontrarysigns,sothatthequantitywemeantorejectisfirstannihilated;or,lastly,thatfromoppositesidesequalquantitiesaresubducted。Andthereforetogetridofquantitiesbythereceivedprinciplesoffluxionsorofdifferencesisneithergoodgeometrynorgoodlogic。Whentheaugmentsvanish,thevelocitiesalsovanish。Thevelocitiesorfluxionsaresaidtobeprimoandultimo,astheaugmentsnascentandevanescent。Takethereforetheratiooftheevanescentquantities,itisthesamewiththatofthefluxions。Itwillthereforeanswerallintentsaswell。Whythenarefluxionsintroduced?Isitnottoshunorrathertopalliatetheuseofquantitiesinfinitelysmall?Butwehavenonotionwherebytoconceiveandmeasurevariousdegreesofvelocitybesidesspaceandtime;or,whenthetimesaregiven,besidesspacealone。Wehaveevennonotionofvelocityprescindedfromtimeandspace。Whenthereforeapointissupposedtomoveingiventimes,wehavenonotionofgreaterorlesservelocities,orofproportionsbetweenvelocities,butonlyoflongerandshorterlines,andofproportionsbetweensuchlinesgeneratedinequalpartsoftime。 31。Apointmaybethelimitofaline:alinemaybethelimitofasurface:amomentmayterminatetime。Buthowcanweconceiveavelocitybythehelpofsuchlimits?Itnecessarilyimpliesbothtimeandspace,andcannotbeconceivedwithoutthem。Andifthevelocitiesofnascentandevanescentquantities,i。e。abstractedfromtimeandspace,maynotbecomprehended,howcanwecomprehendanddemonstratetheirproportions;orconsidertheirrationesprimaeandultimae? For,toconsidertheproportionorratioofthingsimpliesthatsuchthingshavemagnitude;thatsuchtheirmagnitudesmaybemeasured,andtheirrelationstoeachotherknown。But,asthereisnomeasureofvelocityexcepttimeandspace,theproportionofvelocitiesbeingonlycompoundedofthedirectproportionofthespaces,andthereciprocalproportionofthetimes;dothitnotfollowthattotalkofinvestigating,obtaining,andconsideringtheproportionsofvelocities,exclusivelyoftimeandspace,istotalkunintelligibly? 32。Butyouwillsaythat,intheuseandapplicationoffluxions,mendonotoverstraintheirfacultiestoapreciseconceptionoftheabove-mentionedvelocities,increments,infinitesimals,oranyothersuch-likeideasofanaturesonice,subtile,andevanescent。Andthereforeyouwillperhapsmaintainthatproblemsmaybesolvedwithoutthoseinconceivablesuppositions;andthat,consequently,thedoctrineoffluxions,astothepracticalpart,standsclearofallsuchdifficulties。Ianswerthatifintheuseorapplicationofthismethodthosedifficultandobscurepointsarenotattendedto,theyareneverthelesssupposed。Theyarethefoundationsonwhichthemodernsbuild,theprinciplesonwhichtheyproceed,insolvingproblemsanddiscoveringtheorems。Itiswiththemethodoffluxionsaswithallothermethods,whichpresupposetheirrespectiveprinciplesandaregroundedthereon;althoughtherulesmaybepractisedbymenwhoneitherattendto,norperhapsknowtheprinciples。Inlikemanner,therefore,asasailormaypracticallyapplycertainrulesderivedfromastronomyandgeometry,theprincipleswhereofhedothnotunderstand;andasanyordinarymanmaysolvediversnumericalquestions,bythevulgarrulesandoperationsofarithmetic,whichheperformsandapplieswithoutknowingthereasonsofthem:evensoitcannotbedeniedthatyoumayapplytherulesofthefluxionarymethod:youmaycompareandreduceparticularcasestogeneralforms:youmayoperateandcomputeandsolveproblemsthereby,notonlywithoutanactualattentionto,oranactualknowledgeof,thegroundsofthatmethod,andtheprincipleswhereonitdepends,andwhenceitisdeduced,butevenwithouthavingeverconsideredorcomprehendedthem。 33。Butthenitmustberememberedthatinsuchcase,althoughyoumaypassforanartist,computist,oranalyst,yetyoumaynotbejustlyesteemedamanofscienceanddemonstration。Norshouldanyman,invirtueofbeingconversantinsuchobscureanalytics,imaginehisrationalfacultiestobemoreimprovedthanthoseofothermenwhichhavebeenexercisedinadifferentmannerandondifferentsubjects;muchlesserecthimselfintoajudgeandanoracleconcerningmattersthathavenosortofconnexionwithordependenceonthosespecies,symbols,orsigns,inthemanagementwhereofheissoconversantandexpert。Asyou,whoareaskilfulcomputistoranalyst,maynotthereforebedeemedskilfulinanatomy;orviceversa,asamanwhocandissectwithartmay,nevertheless,beignorantinyourartofcomputing:evensoyoumayboth,notwithstandingyourpeculiarskillinyourrespectivearts,bealikeunqualifiedtodecideuponlogic,ormetaphysics,orethics,orreligion。Andthiswouldbetrue,evenadmittingthatyouunderstoodyourownprinciplesandcoulddemonstratethem。 34。Ifitissaidthatfluxionsmaybeexpoundedorexpressedbyfinitelinesproportionaltothem;whichfinitelines,astheymaybedistinctlyconceivedandknownandreasonedupon,sotheymaybesubstitutedforthefluxions,andtheirmutualrelationsorproportionsbeconsideredastheproportionsoffluxions-bywhichmeansthedoctrinebecomesclearanduseful。Ianswerthatif,inordertoarriveatthesefinitelinesproportionaltothefluxions,therebecertainstepsmadeuseofwhichareobscureandinconceivable,bethosefinitelinesthemselveseversoclearlyconceived,itmustneverthelessbeacknowledgedthatyourproceedingisnotclearnoryourmethodscientific。Forinstance,itissupposedthatABbeingtheabscissa,BCtheordinate,andVCHatangentofthecurveAC,BborCEtheincrementoftheabscissa,Ectheincrementoftheordinate,whichproducedmeetsVHinthepointTandCctheincrementofthecurve。TherightlineCcbeingproducedtoK,thereareformedthreesmalltriangles,therectilinearCEc,themixtilinearCEc,andtherectilineartriangleCET。Itisevidentthatthesethreetrianglesaredifferentfromeachother,therectilinearCEcbeinglessthanthemixtilinearCEc,whosesidesarethethreeincrementsabovementioned,andthisstilllessthanthetriangleCET。ItissupposedthattheordinatebcmovesintotheplaceBC,sothatthepointciscoincidentwiththepointC;andtherightlineCK,andconsequentlythecurveCc,iscoincidentwiththetangentCH。InwhichcasethemixtilinearevanescenttriangleCEcwill,initslastform,besimilartothetriangleCET:anditsevanescentsidesCE,EcandCc,willbeproportionaltoCE,ETandCT,thesidesofthetriangleCET。AndthereforeitisconcludedthatthefluxionsofthelinesAB,BC,andAC,beinginthelastratiooftheirevanescentincrements,areproportionaltothesidesofthetriangleCET,or,whichisallone,ofthetriangleVBCsimilarthereunto。[`Introd。adQuadraturamCurvarum。’]Itisparticularlyremarkedandinsistedonbythegreatauthor,thatthepointsCandcmustnotbedistantonefromanother,byanytheleastintervalwhatsoever:butthat,inordertofindtheultimateproportionsofthelinesCE,Ec,andCc(i。e。theproportionsofthefluxionsorvelocities)expressedbythefinitesidesofthetriangleVBC,thepointsCandcmustbeaccuratelycoincident,i。e。oneandthesame。Apointthereforeisconsideredasatriangle,oratriangleissupposedtobeformedinapoint。Whichtoconceiveseemsquiteimpossible。Yetsometherearewho,thoughtheyshrinkatallothermysteries,makenodifficultyoftheirown,whostrainatagnatandswallowacamel。 35。Iknownotwhetheritbeworthwhiletoobserve,thatpossiblysomemenmayhopetooperatebysymbolsandsuppositions,insuchsortastoavoidtheuseoffluxions,momentums,andinfinitesimals,afterthefollowingmanner。Supposextobeoneabscissaofacurve,andzanotherabscissaofthesamecurve。Supposealsothattherespectiveareasarexxxandzzz:andthatz-xistheincrementoftheabscissa,andzzz-xxxtheincrementofthearea,withoutconsideringhowgreatorhowsmallthoseincrementsmaybe。Dividenowzzz-xxxbyz-x,andthequotientwillbezz+zx+xx:and,supposingthatzandxareequal,thesamequotientwillbe3xx,whichinthatcaseistheordinate,whichthereforemaybethusobtainedindependentlyoffluxionsandinfinitesimals。Buthereinisadirectfallacy: forinthefirstplace,itissupposedthattheabscissaezandxareunequal,withoutsuchsuppositionnoonestepcouldhavebeenmade;andinthesecondplace,itissupposedtheyareequal;whichisamanifestinconsistency,andamountstothesamethingthathathbeenbeforeconsidered。[Sect。15。]Andthereisindeedreasontoapprehendthatallattemptsforsettingtheabstruseandfinegeometryonarightfoundation,andavoidingthedoctrineofvelocities,momentums,&;c。 willbefoundimpracticable,tillsuchtimeastheobjectandtheendofgeometryarebetterunderstoodthanhithertotheyseemtohavebeen。Thegreatauthorofthemethodoffluxionsfeltthisdifficulty,andthereforehegaveintothoseniceabstractionsandgeometricalmetaphysicswithoutwhichhesawnothingcouldbedoneonthereceivedprinciples:andwhatinthewayofdemonstrationhehathdonewiththemthereaderwilljudge。 Itmust,indeed,beacknowledgedthatheusedfluxions,likethescaffoldofabuilding,asthingstobelaidasideorgotridofassoonasfinitelineswerefoundproportionaltothem。Butthenthesefiniteexponentsarefoundbythehelpoffluxions。Whateverthereforeisgotbysuchexponentsandproportionsistobeascribedtofluxions:whichmustthereforebepreviouslyunderstood。Andwhatarethesefluxions?Thevelocitiesofevanescentincrements。Andwhatarethesesameevanescentincrements?Theyareneitherfinitequantitiesnorquantitiesinfinitelysmall,noryetnothing。Maywenotcallthemtheghostsofdepartedquantities? 36。Mentoooftenimposeonthemselvesandothersasiftheyconceivedandunderstoodthingsexpressedbysigns,whenintruththeyhavenoidea,saveonlyoftheverysignsthemselves。Andtherearesomegroundstoapprehendthatthismaybethepresentcase。Thevelocitiesofevanescentornascentquantitiesaresupposedtobeexpressed,bothbyfinitelinesofadeterminatemagnitude,andbyalgebraicalnotesorsigns:butIsuspectthatmanywho,perhapsneverhavingexaminedthemattertakeitforgranted,would,uponanarrowscrutiny,finditimpossibletoframeanyideaornotionwhatsoeverofthosevelocities,exclusiveofsuchfinitequantitiesandsigns。SupposethelineKPdescribedbythemotionofapointcontinuallyaccelerated,andthatinequalparticlesoftimetheunequalpartsKL,LM,MN,NO,&;c。aregenerated。Supposealsothata,b,c,d,e,&;c。denotethevelocitiesofthegeneratingpoint,attheseveralperiodsofthepartsorincrementssogenerated。Itiseasytoobservethattheseincrementsareeachproportionaltothesumofthevelocitieswithwhichitisdescribed:that,consequently,theseveralsumsofthevelocities,generatedinequalpartsoftime,maybesetforthbytherespectivelinesKL,LM,MN,&;c。 generatedinthesametimes。Itislikewiseaneasymattertosay,thatthelastvelocitygeneratedinthefirstparticleoftimemaybeexpressedbythesymbola,thelastinthesecondbyb,thelastinthethirdbyc,andsoon:thataisthevelocityofLMinstatunascenti,andb,c,d,e,&;c。 arethevelocitiesoftheincrementsMN,NO,OP,&;c。 intheirrespectivenascentestates。Youmayproceedandconsiderthesevelocitiesthemselvesasflowingorincreasingquantities,takingthevelocitiesofthevelocities,andthevelocitiesofthevelocitiesofthevelocities,i。e。thefirst,second,third&;c。velocitiesadinfinitum: whichsucceedingseriesofvelocitiesmaybethusexpressed,a,b-a,c-2b+a,d-3c+3b-a&;c。whichyoumaycallbythenamesofthefirst,second,third,fourthfluxions。AndforanapterexpressionyoumaydenotethevariableflowinglineKL,KM,KN,&;c。bytheletterx;andthefirstfluxionsby,thesecondby,thethirdby,andsoonadinfinitum。 37。Nothingiseasierthantoassignnames,signs,orexpressionstothesefluxions;anditisnotdifficulttocomputeandoperatebymeansofsuchsigns。Butitwillbefoundmuchmoredifficulttoomitthesignsandyetretaininourmindsthethingswhichwesupposetobesignifiedbythem。Toconsidertheexponents,whethergeometrical,oralgebraical,orfluxionary,isnodifficultmatter。Buttoformapreciseideaofathirdvelocityforinstance,initselfandbyitself,Hocopus,hiclabor。Norindeedisitaneasypointtoformaclearanddistinctideaofanyvelocityatall,exclusiveofandprescindingfromalllengthoftimeandspace;asalsofromallnotes,signs,orsymbolswhatsoever。This,ifImaybeallowedtojudgeofothersbymyself,isimpossible。Tomeitseemsevidentthatmeasuresandsignsareabsolutelynecessaryinordertoconceiveorreasonaboutvelocities;andthatconsequently,whenwethinktoconceivethevelocitiessimplyandinthemselves,wearedeludedbyvainabstractions。 38。Itmayperhapsbethoughtbysomeaneasiermethodofconceivingfluxionstosupposethemthevelocitieswherewiththeinfinitesimaldifferencesaregenerated。Sothatthefirstfluxionsshallbethevelocitiesofthefirstdifferences,thesecondthevelocitiesoftheseconddifferences,thethirdfluxionsthevelocitiesofthethirddifferences,andsoonadinfinitum。But,nottomentiontheinsurmountabledifficultyofadmittingorconceivinginfinitesimals,andinfinitesimalsofinfinitesimals,&;c。,itisevidentthatthisnotionoffluxionswouldnotconsistwiththegreatauthor’sview;whoheldthattheminutestquantityoughtnottobeneglected,thatthereforethedoctrineofinfinitesimaldifferenceswasnottobeadmittedingeometry,andwhoplainlyappearstohaveintroducedtheuseofvelocitiesorfluxions,onpurposetoexcludeordowithoutthem。 39。Toothersitmaypossiblyseemthatweshouldformajusterideaoffluxionsbyassumingthefinite,unequal,isochronalincrementsKL,LM,MN,&;c。,andconsideringtheminstatunascenti,alsotheirincrementsinstatunascenti,andthenascentincrementsofthoseincrements,andsoon,supposingthefirstnascentincrementstobeproportionaltothefirstfluxionsorvelocities,thenascentincrementsofthoseincrementstobeproportionaltothesecondfluxions,thethirdnascentincrementstobeproportionaltothethirdfluxions,andsoonwards。And,asthefirstfluxionsarethevelocitiesofthefirstnascentincrements,sothesecondfluxionsmaybeconceivedtobethevelocitiesofthesecondnascentincrements,ratherthanthevelocitiesofvelocities。Butwhichmeanstheanalogyoffluxionsmayseembetterpreserved,andthenotionrenderedmoreintelligible。 40。Andindeeditshouldseemthatinthewayofobtainingthesecondorthirdfluxionofanequationthegivenfluxionswereconsideredratherasincrementsthanvelocities。Buttheconsideringthemsometimesinonesense,sometimesinanother,onewhileinthemselves,anotherintheirexponents,seemstohaveoccasionednosmallshareofthatconfusionandobscuritywhicharefoundinthedoctrineoffluxions。 Itmayseemthereforethatthenotionmightbestillmended,andthatinsteadoffluxionsoffluxions,offluxionsoffluxionsoffluxions,andinsteadofsecond,third,orfourth,&;c。fluxionsofagivenquantity,itmightbemoreconsistentandlessliabletoexceptiontosay,thefluxionofthefirstnascentincrement,i。e。thesecondfluxion;thefluxionofthesecondnascentincrementi。e。thethirdfluxion;thefluxionofthethirdnascentincrement,i。e。thefourthfluxion-whichfluxionsareconceivedrespectivelyproportional,eachtothenascentprincipleoftheincrementsucceedingthatwhereofitisthefluxion。 41。ForthemoredistinctconceptionofallwhichitmaybeconsideredthatifthefiniteincrementLM[Seetheforegoingschemeinsect。36。]bedividedintotheisochronalpartsLm,mn,no,oM;andtheincrementMNdividedintothepartsMp,pq,qr,rNisochronaltotheformer;asthewholeincrementsLM,MNareproportionaltothesumsoftheirdescribingvelocities,evensothehomologousparticlesLm,Mparealsoproportionaltotherespectiveacceleratedvelocitieswithwhichtheyaredescribed。And,asthevelocitywithwhichMpisgenerated,exceedsthatwithwhichLmwasgenerated,evensotheparticleMpexceedstheparticleLm。Andingeneral,astheisochronalvelocitiesdescribingtheparticlesofMNexceedtheisochronalvelocitiesdescribingtheparticlesofLM,evensotheparticlesoftheformerexceedthecorrespondentparticlesofthelatter。 Andsothiswillhold,bethesaidparticleseversosmall。MNthereforewillexceedLMiftheyarebothtakenintheirnascentstates:andthatexcesswillbeproportionaltotheexcessofthevelocitybabovethevelocitya。Hencewemayseethatthislastaccountoffluxionscomes,intheupshot,tothesamethingwiththefirst。[Sect。 36。] 42。But,notwithstandingwhathathbeensaid,itmuststillbeacknowledgedthatthefiniteparticlesLmorMp,thoughtakeneversosmall,arenotproportionaltothevelocitiesaandb;buteachtoaseriesofvelocitieschangingeverymoment,orwhichisthesamething,toanacceleratedvelocity,bywhichitisgeneratedduringacertainminuteparticleoftime:thatthenascentbeginningsorevanescentendingsoffinitequantities,whichareproducedinmomentsorinfinitelysmallpartsoftime,arealoneproportionaltogivenvelocities: thattherefore,inordertoconceivethefirstfluxions,wemustconceivetimedividedintomoments,incrementsgeneratedinthosemoments,andvelocitiesproportionaltothoseincrements:that,inordertoconceivesecondandthirdfluxions,wemustsupposethatthenascentprinciplesormomentaneousincrementshavethemselvesalsoothermomentaneousincrements,whichareproportionaltotheirrespectivegeneratingvelocities:thatthevelocitiesofthesesecondmomentaneousincrementsaresecondfluxions:thoseoftheirnascentmomentaneousincrementsthirdfluxions。Andsoonadinfinitum。 43。Bysubductingtheincrementgeneratedinthefirstmomentfromthatgeneratedinthesecond,wegettheincrementofanincrement。Andbysubductingthevelocitygeneratinginthefirstmomentfromthatgeneratinginthesecond,wegetafluxionofafluxion。Inlikemanner,bysubductingthedifferenceofthevelocitiesgeneratinginthetwofirstmomentsfromtheexcessofthevelocityinthethirdabovethatinthesecondmoment,weobtainthethirdfluxion。Andafterthesameanalogywemayproceedtofourth,fifth,sixthfluxions&;c。Andifwecallthevelocitiesofthefirst,second,third,fourthmoments,a,b,c,d,theseriesoffluxionswillbeasabove,a,b-a,c-2b+a,d-3c+3b-a,adinfinitum,i。e。,,,,adinfinitum。 44。Thusfluxionsmaybeconceivedinsundrylightsandshapes,whichseemallequallydifficulttoconceive。And,indeed,asitisimpossibletoconceivevelocitywithouttimeorspace,withouteitherfinitelengthorfiniteduration,[Sect。31]itmustseemabovethepowersofmentocomprehendeventhefirstfluxions。Andifthefirstareincomprehensible,whatshallwesayofthesecondandthirdfluxions,&;c。?Hewhocanconceivethebeginningofabeginning,ortheendofanend,somewhatbeforethefirstorafterthelast,maybeperhapssharpsightedenoughtoconceivethesethings。Butmostmenwill,Ibelieve,finditimpossibletounderstandtheminanysensewhatever。 45。Onewouldthinkthatmencouldnotspeaktooexactlyonsoniceasubject。Andyet,aswasbeforehinted,wemayoftenobservethattheexponentsoffluxions,ornotesrepresentingfluxionsarecompoundedwiththefluxionsthemselves。Isnotthisthecasewhen,justafterthefluxionsofflowingquantitiesweresaidtobetheceleritiesoftheirincreasing,andthesecondfluxionstobethemutationsofthefirstfluxionsorcelerities,wearetoldthat[`DeQuadraturaCurvarum。’]representsaseriesofquantitieswhereofeachsubsequentquantityisthefluxionofthepreceding:andeachforegoingisafluentquantityhavingthefollowingoneforitsfluxion? 46。Diversseriesofquantitiesandexpressions,geometricalandalgebraical,maybeeasilyconceived,inlines,insurfaces,inspecies,tobecontinuedwithoutendorlimit。Butitwillnotbefoundsoeasytoconceiveaseries,eitherofmerevelocitiesorofmerenascentincrements,distincttherefromandcorrespondingthereunto。Someperhapsmaybeledtothinktheauthorintendedaseriesofordinates,whereineachordinatewasthefluxionoftheprecedingandfluentofthefollowing,i。e。thatthefluxionofoneordinatewasitselftheordinateofanothercurve;andthefluxionofthislastordinatewastheordinateofyetanothercurve;andsoonadinfinitum。Butwhocanconceivehowthefluxion(whethervelocityornascentincrement)ofanordinateshouldbeitselfanordinate?Ormorethanthateachprecedingquantityorfluentisrelatedtoitssubsequentorfluxion,astheareaofacurvilinearfiguretoitsordinate;agreeablytowhattheauthorremarks,thateachprecedingquantityinsuchseriesisastheareaofacurvilinearfigure,whereoftheabscissaisz,andtheordinateisthefollowingquantity? 47。Uponthewholeitappearsthattheceleritiesaredismissed,andinsteadthereofareasandordinatesareintroduced。 But,howeverexpedientsuchanalogiesorsuchexpressionsmaybefoundforfacilitatingthemodernquadratures,yetweshallnotfindanylightgivenustherebyintotheoriginalrealnatureoffluxions;orthatweareenabledtoframefromthencejustideasoffluxionsconsideredinthemselves。 Inallthisthegeneralultimatedriftoftheauthorisveryclear,buthisprinciplesareobscure。Butperhapsthosetheoriesofthegreatauthorarenotminutelyconsideredorcanvassedbyhisdisciples;whoseemeager,aswasbeforehinted,rathertooperatethantoknow,rathertoapplyhisrulesandhisformsthantounderstandhisprinciplesandenterintohisnotions。Itisneverthelesscertainthat,inordertofollowhiminhisquadratures,theymustfindfluentsfromfluxions;andinordertothis,theymustknowtofindfluxionsfromfluents;andinordertofindfluxions,theymustfirstknowwhatfluxionsare。Otherwisetheyproceedwithoutclearnessandwithoutscience。Thusthedirectmethodprecedestheinverse,andtheknowledgeoftheprinciplesissupposedinboth。Butasforoperatingaccordingtorules,andbythehelpofgeneralforms,whereoftheoriginalprinciplesandreasonsarenotunderstood,thisistobeesteemedmerelytechnical。Betheprinciplesthereforeeversoabstruseandmetaphysical,theymustbestudiedbywhoeverwouldcomprehendthedoctrineoffluxions。 Norcananygeometricianhavearighttoapplytherulesofthegreatauthor,withoutfirstconsideringhismetaphysicalnotionswhencetheywerederived。 These,howevernecessarysoeverinordertoscience,whichcanneverbeobtainedwithoutaprecise,clear,andaccurateconceptionoftheprinciples-areneverthelessbyseveralcarelesslypassedover;whiletheexpressionsalonearedweltonandconsideredandtreatedwithgreatskillandmanagement,thencetoobtainotherexpressionsbymethodssuspiciousandindirect(tosaytheleast)ifconsideredinthemselves,howeverrecommendedbyInductionandAuthority;twomotiveswhichareacknowledgedsufficienttobegetarationalfaithandmoralpersuasion,butnothinghigher。 48。Youmaypossiblyhopetoevadetheforceofallthathathbeensaid,andtoscreenfalseprinciplesandinconsistentreasonings,byageneralpretencethattheseobjectionsandremarksaremetaphysical。Butthisisavainpretence。Fortheplainsenseandtruthofwhatisadvancedintheforegoingremarks,Iappealtotheunderstandingofeveryunprejudicedintelligentreader。TothesameIappeal,whetherthepointsremarkeduponarenotmostincomprehensiblemetaphysics。Andmetaphysicsnotofmine,butyourown。Iwouldnotbeunderstoodtoinferthatyournotionsarefalseorvainbecausetheyaremetaphysical。Nothingiseithertrueoffalseforthatreason。Whetherapointbecalledmetaphysicalornoavailslittle。Thequestionis,whetheritbeclearorobscure,rightorwrong,wellorilldeduced? 49。Althoughmomentaneousincrements,nascentandevanescentquantities,fluxionsandinfinitesimalsofalldegreesareintruthsuchshadowyentities,sodifficulttoimagineorconceivedistinctly,that(tosaytheleast)theycannotbeadmittedasprinciplesorobjectsofclearandaccuratescience;andalthoughthisobscurityandincomprehensibilityofyourmetaphysicshadbeenalonesufficienttoallayyourpretensionstoevidence;yetithath,ifImistakenot,beenfurthershewn,thatyourinferencesarenomorejustthanyourconceptionsareclear,andthatyourlogicsareasexceptionableasyourmetaphysics。Itwouldseem,therefore,uponthewhole,thatyourconclusionsarenotattainedbyjustreasoningfromclearprinciples:consequently,thattheemploymentofmodernanalysts,howeverusefulinmathematicalcalculationsandconstructions,dothnothabituateandqualifythemindtoapprehendclearlyandinferjustly;and,consequently,thatyouhavenoright,invirtueofsuchhabits,todictateoutofyourpropersphere,beyondwhichyourjudgmentistopassfornomorethanthatofothermen。 50。OfalongtimeIhavesuspectedthatthesemodernanalyticswerenotscientifical,andgavesomehintsthereoftothepublicabouttwenty-fiveyearsago。Sincewhichtime,Ihavebeendivertedbyotheroccupations,andimaginedImightemploymyselfbetterthanindeducingandlayingtogethermythoughtsonsoniceasubject。AndthoughoflateIhavebeencalledupontomakegoodmysuggestions;yet,asthepersonwhomadethiscalldothnotappeartothinkmaturelyenoughtounderstandeitherthosemetaphysicswhichhewouldrefute,ormathematicswhichhewouldpatronize,Ishouldhavesparedmyselfthetroubleofwritingforhisconviction。NorshouldInowhavetroubledyouormyselfwiththisaddress,aftersolonganintermissionofthesestudies,wereitnottoprevent,sofarasIamable,yourimposingonyourselfandothersinmattersofmuchhighermomentandconcern。And,totheendthatyoumaymoreclearlycomprehendtheforceanddesignoftheforegoingremarks,andpursuethemstillfartherinyourownmeditations,IshallsubjointhefollowingQueries。 Query1。Whethertheobjectofgeometrybenottheproportionsofassignableextensions?Andwhethertherebeanyneedofconsideringquantitieseitherinfinitelygreatorinfinitelysmall? Qu。2。Whethertheendofgeometrybenottomeasureassignablefiniteextension?Andwhetherthispracticalviewdidnotfirstputmenonthestudyofgeometry? Qu。3。Whetherthemistakingtheobjectandendofgeometryhathnotcreatedneedlessdifficulties,andwrongpursuitsinthatscience? Qu。4。Whethermenmayproperlybesaidtoproceedinascientificmethod,withoutclearlyconceivingtheobjecttheyareconversantabout,theendproposed,andthemethodbywhichitispursued? Qu。5。Whetheritdothnotsuffice,thateveryassignablenumberofpartsmaybecontainedinsomeassignablemagnitude? Andwhetheritbenotunnecessary,aswellasabsurd,tosupposethatfiniteextensionisinfinitelydivisible? Qu。6。Whetherthediagramsinageometricaldemonstrationarenottobeconsideredassignsofallpossiblefinitefigures,ofallsensibleandimaginableextensionsormagnitudesofthesamekind? Qu。7。Whetheritbepossibletofreegeometryfrominsuperabledifficultiesandabsurdities,solongaseithertheabstractgeneralideaofextension,orabsoluteexternalextensionbesupposeditstrueobject? Qu。8。Whetherthenotionsofabsolutetime,absoluteplace,andabsolutemotionbenotmostabstractedlymetaphysical? Whetheritbepossibleforustomeasure,compute,orknowthem? Qu。9。Whethermathematiciansdonotengagethemselvesindisputesandparadoxesconcerningwhattheyneitherdonorcanconceive?Andwhetherthedoctrineofforcesbenotasufficientproofofthis?[SeeaLatintreatise,`DeMotu,’publishedatLondonintheyear1721。] Qu。10。Whetheringeometryitmaynotsufficetoconsiderassignablefinitemagnitude,withoutconcerningourselveswithinfinity?Andwhetheritwouldnotberightertomeasurelargepolygonshavingfinitesides,insteadofcurves,thantosupposecurvesarepolygonsofinfinitesimalsides,asuppositionneithertruenorconceivable? Qu。11。Whethermanypointswhicharenotreadilyassentedtoarenotneverthelesstrue?Andwhoseinthetwofollowingqueriesmaynotbeofthatnumber? Qu。12。Whetheritbepossiblethatweshouldhavehadanideaornotionofextensionpriortomotion?Orwhether,ifamanhadneverperceivedmotion,hewouldeverhaveknownorconceivedonethingtobedistantfromanother? Qu。13。Whethergeometricalquantityhathco-existentparts?Andwhetherallquantitybenotinafluxaswellastimeandmotion? Qu。14。WhetherextensioncanbesupposedanattributeofaBeingimmutableandeternal? Qu。15。Whethertodeclineexaminingtheprinciples,andunravellingthemethodsusedinmathematicswouldnotshewabigotryinmathematicians? Qu。16。Whethercertainmaximsdonotpasscurrentamonganalystswhichareshockingtogoodsense?Andwhetherthecommonassumption,thatafinitequantitydividedbynothingisinfinite,benotofthisnumber? Qu。17。Whethertheconsideringgeometricaldiagramsabsolutelyorinthemselves,ratherthanasrepresentativesofallassignablemagnitudesorfiguresofthesamekind,benotaprinciplecauseofthesupposingfiniteextensioninfinitelydivisible;andofallthedifficultiesandabsurditiesconsequentthereupon? Qu。18。Whether,fromgeometricalpropositionsbeinggeneral,andthelinesindiagramsbeingthereforegeneralsubstitutesorrepresentatives,itdothnotfollowthatwemaynotlimitorconsiderthenumberofpartsintowhichsuchparticularlinesaredivisible? Qu。19。Whenitissaidorimplied,thatsuchacertainlinedelineatedonpapercontainsmorethananyassignablenumberofparts,whetheranymoreintruthoughttobeunderstood,thanthatitisasignindifferentlyrepresentingallfinitelines,betheyeversogreat。Inwhichrelativecapacityitcontains,i。e。standsformorethananyassignablenumberofparts?Andwhetheritbenotaltogetherabsurdtosupposeafiniteline,consideredinitselforinitsownpositivenature,shouldcontainaninfinitenumberofparts? Qu。20。Whetherallargumentsfortheinfinitedivisibilityoffiniteextensiondonotsupposeandimply,eithergeneralabstractideas,orabsoluteexternalextensiontobetheobjectofgeometry? And,therefore,whether,alongwiththosesuppositions,suchargumentsalsodonotceaseandvanish? Qu。21。Whetherthesupposedinfinitedivisibilityoffiniteextensionhathnotbeenasnaretomathematiciansandathornintheirsides?Andwhetheraquantityinfinitelydiminishedandaquantityinfinitelysmallarenotthesamething? Qu。22。Whetheritbenecessarytoconsidervelocitiesofnascentorevanescentquantities,ormoments,orinfinitesimals? Andwhethertheintroducingofthingssoinconceivablebenotareproachtomathematics? Qu。23。Whetherinconsistenciescanbetruths?Whetherpointsrepugnantandabsurdaretobeadmitteduponanysubjects,orinanyscience?Andwhethertheuseofinfinitesoughttobeallowedasasufficientpretextandapologyfortheadmittingofsuchpointsingeometry? Qu。24。Whetheraquantitybenotproperlysaidtobeknown,whenweknowitsproportiontogivenquantities?Andwhetherthisproportioncanbeknownbutbyexpressionsorexponents,eithergeometrical,algebraical,orarithmetical?Andwhetherexpressionsinlinesorspeciescanbeusefulbutsofarforthastheyarereducibletonumbers? Qu。25。Whetherthefindingoutproperexpressionsornotationsofquantitybenotthemostgeneralcharacterandtendencyofthemathematics?Andarithmeticaloperationthatwhichlimitsanddefinestheiruse? Qu。26。Whethermathematicianshavesufficientlyconsideredtheanalogyanduseofsigns?Andhowfarthespecificlimitednatureofthingscorrespondsthereto? Qu。27。Whetherbecause,instatingageneralcaseofpurealgebra,weareatfulllibertytomakeacharacterdenoteeitherapositiveoranegativequantity,ornothingatall,wemaytherefore,inageometricalcase,limitedbyhypothesesandreasoningsfromparticularpropertiesandrelationsoffigures,claimthesamelicence? Qu。28。Whethertheshiftingofthehypothesis,or(aswemaycallit)thefallaciasuppositionisbenotasophismthatfarandwideinfectsthemodernreasonings,bothinthemechanicalphilosophyandintheabstruseandfinegeometry? Qu。29。Whetherwecanformanideaornotionofvelocitydistinctfromandexclusiveofitsmeasures,aswecanofheatdistinctfromandexclusiveofthedegreesonthethermometerbywhichitismeasured?Andwhetherthisbenotsupposedinthereasoningsofmodernanalysts? Qu。30。Whethermotioncanbeconceivedinapointofspace?Andifmotioncannot,whethervelocitycan?Andifnot,whetherafirstorlastvelocitycanbeconceivedinamerelimit,eitherinitialorfinal,ofthedescribedspace? Qu。31。Wheretherearenoincrements,whethertherecanbeanyratioofincrements?Whethernothingscanbeconsideredasproportionaltorealquantities?Orwhethertotalkoftheirproportionsbenottotalknonsense?Alsoinwhatsensewearetounderstandtheproportionofasurfacetoaline,ofanareatoanordinate? Andwhetherspeciesornumbers,thoughproperlyexpressingquantitieswhicharenothomogeneous,mayyetbesaidtoexpresstheirproportiontoeachother? Qu。32。Whetherifallassignablecirclesmaybesquared,thecircleisnot,toallintentsandpurposes,squaredaswellastheparabola?Ofwhetheraparabolicareacaninfactbemeasuredmoreaccuratelythanacircular? Qu。33。Whetheritwouldnotberightertoapproximatefairlythantoendeavourataccuracybysophisms? Qu。34。Whetheritwouldnotbemoredecenttoproceedbytrialsandinductions,thantopretendtodemonstratebyfalseprinciples? Qu。35。Whethertherebenotawayofarrivingattruth,althoughtheprinciplesarenotscientific,northereasoningjust?Andwhethersuchawayoughttobecalledaknackorascience? Qu。36。Whethertherecanbescienceoftheconclusionwherethereisnotevidenceoftheprinciples?Andwhetheramancanhaveevidenceoftheprincipleswithoutunderstandingthem?Andtherefore,whetherthemathematiciansofthepresentageactlikemenofscience,intakingsomuchmorepainstoapplytheirprinciplesthantounderstandthem? Qu。37。Whetherthegreatestgeniuswrestlingwithfalseprinciplesmaynotbefoiled?Andwhetheraccuratequadraturescanbeobtainedwithoutnewpostulataorassumptions?Andifnot,whetherthosewhichareintelligibleandconsistentoughtnottobepreferredtothecontrary?Seesect。28and29。 Qu。38。Whethertediouscalculationsinalgebraandfluxionsbethelikeliestmethodtoimprovethemind?Andwhethermen’sbeingaccustomedtoreasonaltogetheraboutmathematicalsignsandfiguresdothnotmakethematalosshowtoreasonwithoutthem? Qu。39。Whether,whateverreadinessanalystsacquireinstatingaproblem,orfindingaptexpressionsformathematicalquantities,thesamedothnecessarilyinferaproportionableabilityinconceivingandexpressingothermatters? Qu。40。Whetheritbenotageneralcaseorrule,thatoneandthesamecoefficientdividingequalproductsgivesequalquotients?Andyetwhethersuchcoefficientcanbeinterpretedbyoornothing?Orwhetheranyonewillsaythatiftheequation2o=5obedividedbyo,thequotientsonbothsidesareequal?Whetherthereforeacasemaynotbegeneralwithrespecttoallquantitiesandyetnotextendtonothings,orincludethecaseofnothing?Andwhetherthebringingnothingunderthenotionofquantitymaynothavebetrayedmenintofalsereasoning? Qu。41。Whetherinthemostgeneralreasoningsaboutequalitiesandproportionsmenmaynotdemonstrateaswellasingeometry?Whetherinsuchdemonstrationstheyarenotobligedtothesamestrictreasoningasingeometry?Andwhethersuchtheirreasoningsarenotdeducedfromthesameaxiomswiththoseingeometry?Whetherthereforealgebrabenotastrulyascienceasgeometry? Qu。42。Whethermenmaynotreasoninspeciesaswellasinwords?Whetherthesamerulesoflogicdonotobtaininbothcases?Andwhetherwehavenotarighttoexpectanddemandthesameevidenceinboth? Qu。43。Whetheranalgebraist,fluxionist,geometrician,ordemonstratorofanykindcanexpectindulgenceforobscureprinciplesorincorrectreasonings?Andwhetheranalgebraicalnoteorspeciescanattheendofaprocessbeinterpretedinasensewhichcouldnothavebeensubstitutedforitatthebeginning?Orwhetheranyparticularsuppositioncancomeunderageneralcasewhichdothnotconsistwiththereasoningthereof? Qu。44。Whetherthedifferencebetweenamerecomputerandamanofsciencebenot,thattheonecomputesonprinciplesclearlyconceived,andbyrulesevidentlydemonstrated,whereastheotherdothnot? Qu。45。Whether,althoughgeometrybeascience,andalgebraallowedtobeascience,andtheanalyticalamostexcellentmethod,intheapplication,nevertheless,oftheanalysistogeometry,menmaynothaveadmittedfalseprinciplesandwrongmethodsofreasoning? Qu。46。Whether,althoughalgebraicalreasoningsareadmittedtobeeversojust,whenconfinedtosignsorspeciesasgeneralrepresentativesofquantity,youmaynotneverthelessfallintoerror,if,whenyoulimitthemtostandforparticularthings,youdonotlimityourselftoreasonconsistentlywiththenatureofsuchparticularthings? Andwhethersucherroroughttobeimputedtopurealgebra? Qu。47。Whethertheviewofmodernmathematiciansdothnotratherseemtobethecomingatanexpressionbyartifice,thanatthecomingatsciencebydemonstration? Qu。48。Whethertheremaynotbesoundmetaphysicsaswellasunsound?Soundaswellasunsoundlogic?Andwhetherthemodernanalyticsmaynotbebroughtunderoneofthesedenominations,andwhich? Qu。49。Whethertherebenotreallyaphilosophiaprima,acertaintranscendentalsciencesuperiortoandmoreextensivethanmathematics,whichitmightbehoveourmodernanalystsrathertolearnthandespise? Qu。50。Whether,eversincetherecoveryofmathematicallearning,therehavenotbeenperpetualdisputesandcontroversiesamongthemathematicians?Andwhetherthisdothnotdisparagetheevidenceoftheirmethods? Qu。51。Whetheranythingbutmetaphysicsandlogiccanopentheeyesofmathematiciansandextricatethemoutoftheirdifficulties? Qu。52。Whether,uponthereceivedprinciples,aquantitycanbyanydivisionorsubdivision,thoughcarriedeversofar,bereducedtonothing? Qu。53。Whether,iftheendofgeometrybepractice,andthispracticebemeasuring,andwemeasureonlyassignableextensions,itwillnotfollowthatunlimitedapproximationscompletelyanswertheintentionofgeometry? Qu。54。Whetherthesamethingswhicharenowdonebyinfinitesmaynotbedonebyfinitequantities?Andwhetherthiswouldnotbeagreatrelieftotheimaginationsandunderstandingsofmathematicalmen? Qu。55。Whetherthosephilomathematicalphysicians,anatomists,anddealersintheanimaleconomy,whoadmitthedoctrineoffluxionswithanimplicitfaith,canwithagoodgraceinsultothermenforbelievingwhattheydonotcomprehend? Qu。56。Whetherthecorpuscularian,experimental,andmathematicalphilosophy,somuchcultivatedinthelastage,hathnottoomuchengrossedmen’sattention;somepartwhereofitmighthaveusefullyemployed? Qu。57。Whether,fromthisandotherconcurringcauses,themindsofspeculativemenhavenotbeenbornedownward,tothedebasingandstupifyingofthehigherfaculties?Andwhetherwemaynothenceaccountforthatprevailingnarrownessandbigotryamongmanywhopassformenofscience,theirincapacityforthingsmoral,intellectual,ortheological,theirpronenesstomeasurealltruthsbysenseandexperienceofanimallife? Qu。58。Whetheritbereallyaneffectofthinking,thatthesamemenadmirethegreatauthorforhisfluxions,andderidehimforhisreligion? Qu。59。Ifcertainphilosophicalvirtuosiofthepresentagehavenoreligion,whetheritcanbesaidtobewantoffaith? Qu。60。Whetheritbenotajusterwayofreasoning,torecommendpointsoffaithfromtheireffects,thantodemonstratemathematicalprinciplesbytheirconclusions? Qu。61。Whetheritbenotlessexceptionabletoadmitpointsabovereasonthancontrarytoreason? Qu。62。WhethermysteriesmaynotwithbetterrightbeallowedofinDivineFaiththaninhumanscience? Qu。63。Whethersuchmathematiciansascryoutagainstmysterieshaveeverexaminedtheirownprinciples? Qu。64。Whethermathematicians,whoaresodelicateinreligiouspoints,arestrictlyscrupulousintheirownscience? Whethertheydonotsubmittoauthority,takethingsupontrust,andbelievepointsinconceivable?Whethertheyhavenottheirmysteries,andwhatismore,theirrepugnancesandcontradictions? Qu。65。Whetheritmightnotbecomemenwhoarepuzzledandperplexedabouttheirownprinciples,tojudgewarily,candidly,andmodestlyconcerningothermatters? Qu。66。Whetherthemodernanalyticsdonotfurnishastrongargumentumadhominemagainstthephilomathematicalinfidelsofthesetimes? Qu。67。Whetheritfollowsfromtheabove-mentionedremarks,thataccurateandjustreasoningisthepeculiarcharacterofthepresentage?Andwhetherthemoderngrowthofinfidelitycanbeascribedtoadistinctionsotrulyvaluable?