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？
