aDiscourseaddressedtoanInfidelMathematicianTheAnalyst：aDiscourseaddressedtoanInfidelMathematicianByGeorgeBerkeleyContents1。Mathematicianspresumedtobethegreatmastersofreason。Henceanunduedeferencetotheirdecisionswheretheyhavenorighttodecide。Thisonecauseofinfidelity。
2。Theirprinciplesandmethodstobeexaminedwiththesamefreedomwhichtheyassumewithregardtotheprinciplesandmysteriesofreligion。Inwhatsenseandhowfargeometryistobeallowedanimprovementofthemind。
3。Fluxionsthegreatobjectandemploymentoftheprofoundgeometriciansinthepresentage。Whatthesefluxionsare。
4。Momentsornascentincrementsofflowingquantitiesdifficulttoconceive。Fluxionsofdifferentorders。Secondandthirdfluxionsobscuremysteries。
5。Differences，i。e。incrementsordecrementsinfinitelysmall，usedbyforeignmathematiciansinsteadoffluxionsorvelocitiesofnascentandevanescentincrements。
6。Differencesofvariousorders，i。e。quantitiesinfinitelylessthanquantitiesinfinitelylittle；andinfinitesimalpartsofinfinitesimalsofinfinitesimals，&；c。withoutendorlimit。
7。Mysteriesinfaithunjustlyobjectedagainstbythosewhoadmittheminscience。
8。ModernAnalystssupposedbythemselvestoextendtheirviewsevenbeyondinfinity：deludedbytheirownspeciesorsymbols。
9。Methodforfindingthefluxionofarectangleoftwoindeterminatequantities，shewedtobeillegitimateandfalse。
10。Implicitdeferenceofmathematicalmenforthegreatauthoroffluxions。Theirearnestnessrathertogoonfastandfar，thantosetoutwarilyandseetheirwaydistinctly。
11。Momentumsdifficulttocomprehend。Nomiddlequantitytobeadmittedbetweenafinitequantityandnothing，withoutadmittinginfinitesimals。
12。Thefluxionofanypowerofaflowingquantity。
Lemmapremisedinordertoexaminethemethodforfindingsuchfluxion。
13。Theruleforfluxionsofpowersattainedbyunfairreasoning。
14。Theaforesaidreasoningfartherunfolded，andshewedtobeillogical。
15。Notrueconclusiontobejustlydrawnbydirectconsequencefrominconsistentsuppositions。Thesamerulesofrightreasontobeobserved，whethermenargueinsymbolsorinwords。
16。Anhypothesisbeingdestroyed，noconsequenceofsuchhypothesistoberetained。
17。Hardtodistinguishbetweenevanescentincrementsandinfinitesimaldifferences。Fluxionsplacedinvariouslights。Thegreatauthor，itseems，notsatisfiedwithhisownnotions。
18。QuantitiesinfinitelysmallsupposedandrejectedbyLeibnitzandhisfollowers。Noquantity，accordingtothem，greaterorsmallerfortheadditionorsubductionofitsinfinitesimal。
19。Conclusionstobeprovedbytheprinciples，andnotprinciplesbytheconclusions。
20。ThegeometricalAnalystconsideredasalogician；
andhisdiscoveries，notinthemselves，butasderivedfromsuchprinciplesandbysuchinferences。
21。Atangentdrawntotheparabolaaccordingtothecalculusdifferentialis。Truthshewntobetheresultoferror，andhow。
22。ByvirtueofatwofoldmistakeAnalystsarriveattruth，butnotatscience：ignoranthowtheycomeattheirownconclusions。
23。Theconclusionneverevidentoraccurate，invirtueofobscureorinaccuratepremises。Finitequantitiesmightberejectedaswellasinfinitesimals。
24。Theforegoingdoctrinefartherillustrated。
25。Sundryobservationsthereupon。
26。Ordinatefoundfromtheareabymeansofevanescentincrements。
27。Intheforegoingcase，thesupposedevanescentincrementisreallyafinitequantity，destroyedbyanequalquantitywithanoppositesign。
28。Theforegoingcaseputgenerally。Algebraicalexpressionscomparedwiththegeometricalquantities。
29。Correspondentquantitiesalgebraicalandgeometricalequated。Theanalysisshewednottoobtainininfinitesimals，butitmustalsoobtaininfinitequantities。
30。Thegettingridofquantitiesbythereceivedprinciples，whetheroffluxionsorofdifferences，neithergoodgeometrynorgoodlogic。Fluxionsorvelocities，whyintroduced。
31。Velocitiesnottobeabstractedfromtimeandspace：northeirproportionstobeinvestigatedorconsideredexclusivelyoftimeandspace。
32。DifficultandobscurepointsconstitutetheprinciplesofthemodernAnalysis，andarethefoundationonwhichitisbuilt。
33。Therationalfacultieswhetherimprovedbysuchobscureanalytics。
34。Bywhatinconceivablestepsfinitelinesarefoundproportionaltofluxions。Mathematicalinfidelsstrainatagnatandswallowacamel。
35。Fluxionsorinfinitesimalsnottobeavoidedonthereceivedprinciples。Niceabstractionsandgeometricalmetaphysics。
36。Velocitiesofnascentorevanescentquantities，whetherinrealityunderstoodandsignifiedbyfinitelinesandspecies。
37。Signsorexponentsobvious；butfluxionsthemselvesnotso。
38。Fluxions，whetherthevelocitieswithwhichinfinitesimaldifferencesaregenerated。
39。Fluxionsoffluxionsorsecondfluxions，whethertobeconceivedasvelocitiesofvelocities，orratherasvelocitiesofthesecondnascentincrements。
40。Fluxionsconsidered，sometimesinonesense，sometimesinanother；onewhileinthemselves，anotherintheirexponents：
henceconfusionandobscurity。
41。Isochronalincrements，whetherfiniteornascent，proportionaltotheirrespectivevelocities。
42。Timesupposedtobedividedintomoments：incrementsgeneratedinthosemoments：andvelocitiesproportionaltothoseincrements。
43。Fluxions，second，third，fourth，&；c。，whattheyare，howobtained，andhowrepresented。Whatideaofvelocityinamomentoftimeandpointofspace。
44。Fluxionsofallordersinconceivable。
45。Signsorexponentsconfoundedwiththefluxions。
46。Seriesofexpressionsorofnoteseasilycontrived。
Whetheraseriesofmerevelocities，orofmerenascentincrementscorrespondingthereunto，beaseasilyconceived。
47。Celeritiesdismissed，andinsteadthereofordinatesandareasintroduced。Analogiesandexpressionsusefulinthemodernquadratures，mayyetbeuselessforenablingustoconceivefluxions。Norighttoapplytheruleswithoutknowledgeoftheprinciples。
48。MetaphysicsofmodernAnalystsmostincomprehensible。
49。Analystsemployedaboutnotionalshadowyentities。
Theirlogicsasexceptionableastheirmetaphysics。
50。Occasionofthisaddress。Conclusion。Queries。
THEANALYST：
ADiscourseaddressedtoanInfidelMathematician1。ThoughIamastrangertoyourperson，yetIamnot，Sir，astrangertothereputationyouhaveacquiredinthatbranchoflearningwhichhathbeenyourpeculiarstudy；nortotheauthoritythatyouthereforeassumeinthingsforeigntoyourprofession；nortotheabusethatyou，andtoomanymoreofthelikecharacter，areknowntomakeuseofsuchundueauthority，tothemisleadingofunwarypersonsinmattersofthehighestconcernment，andwhereofyourmathematicalknowledgecanbynomeansqualifyyoutobeacompetentjudge。Equityindeedandgoodsensewouldinclineonetodisregardthejudgmentofmen，inpointswhichtheyhavenotconsideredorexamined。Butseveralwhomaketheloudestclaimtothosequalitiesdoneverthelesstheverythingtheywouldseemtodespise，clothingthemselvesintheliveryofothermen’sopinions，andputtingonageneraldeferenceforthejudgmentofyou，Gentlemen，whoarepresumedtobeofallmenthegreatestmastersofreason，tobemostconversantaboutdistinctideas，andnevertotakethingsontrust，butalwaysclearlytoseeyourway，asmenwhoseconstantemploymentisthededucingtruthbythejustestinferencefromthemostevidentprinciples。
Withthisbiasontheirminds，theysubmittoyourdecisionswhereyouhavenorighttodecide。AndthatthisisoneshortwayofmakingInfidels，Iamcrediblyinformed。
2。Whereasthenitissupposedthatyouapprehendmoredistinctly，considermoreclosely，infermorejustly，andconcludemoreaccuratelythanothermen，andthatyouarethereforelessreligiousbecausemorejudicious，IshallclaimtheprivilegeofaFree-thinker；
andtakethelibertytoinquireintotheobject，principles，andmethodofdemonstrationadmittedbythemathematiciansofthepresentage，withthesamefreedomthatyoupresumetotreattheprinciplesandmysteriesofReligion；totheendthatallmenmayseewhatrightyouhavetolead，orwhatencouragementothershavetofollowyou。Ithathbeenanoldremark，thatGeometryisanexcellentLogic。Anditmustbeownedthatwhenthedefinitionsareclear；whenthepostulatacannotberefused，northeaxiomsdenied；whenfromthedistinctcontemplationandcomparisonoffigures，theirpropertiesarederived，byaperpetualwell-connectedchainofconsequences，theobjectsbeingstillkeptinview，andtheattentioneverfixeduponthem；thereisacquiredahabitofreasoning，closeandexactandmethodical：
whichhabitstrengthensandsharpensthemind，andbeingtransferredtoothersubjectsisofgeneraluseintheinquiryaftertruth。Buthowfarthisisthecaseofourgeometricalanalysts，itmaybeworthwhiletoconsider。
3。ThemethodofFluxionsisthegeneralkeybyhelpwhereofthemodernmathematiciansunlockthesecretsofGeometry，andconsequentlyofNature。And，asitisthatwhichhathenabledthemsoremarkablytooutgotheancientsindiscoveringtheoremsandsolvingproblems，theexerciseandapplicationthereofisbecomethemainifnotsoleemploymentofallthosewhointhisagepassforprofoundgeometers。
Butwhetherthismethodbeclearorobscure，consistentorrepugnant，demonstrativeorprecarious，asIshallinquirewiththeutmostimpartiality，soIsubmitmyinquirytoyourownjudgment，andthatofeverycandidreader。Linesaresupposedtobegenerated[`Introd。adQuadraturamCurvarum。’]bythemotionofpoints，planesbythemotionoflines，andsolidsbythemotionofplanes。Andwhereasquantitiesgeneratedinequaltimesaregreaterorlesseraccordingtothegreaterorlesservelocitywherewiththeyincreaseandaregenerated，amethodhathbeenfoundtodeterminequantitiesfromthevelocitiesoftheirgeneratingmotions。Andsuchvelocitiesarecalledfluxions：andthequantitiesgeneratedarecalledflowingquantities。Thesefluxionsaresaidtobenearlyastheincrementsoftheflowingquantities，generatedintheleastequalparticlesoftime；andtobeaccuratelyinthefirstproportionofthenascent，orinthelastoftheevanescentincrements。
Sometimes，insteadofvelocities，themomentaneousincrementsordecrementsofundeterminedflowingquantitiesareconsidered，undertheappellationofmoments。
4。Bymomentswearenottounderstandfiniteparticles。
Thesearesaidnottobemoments，butquantitiesgeneratedfrommoments，whichlastareonlythenascentprinciplesoffinitequantities。Itissaidthattheminutesterrorsarenottobeneglectedinmathematics：thatthefluxionsarecelerities，notproportionaltothefiniteincrements，thougheversosmall；butonlytothemomentsornascentincrements，whereoftheproportionalone，andnotthemagnitude，isconsidered。Andoftheaforesaidfluxionstherebeotherfluxions，whichfluxionsoffluxionsarecalledsecondfluxions。Andthefluxionsofthesesecondfluxionsarecalledthirdfluxions：andsoon，fourth，fifth，sixth，&；c。adinfinitum。
Now，asourSenseisstrainedandpuzzledwiththeperceptionofobjectsextremelyminute，evensotheImagination，whichfacultyderivesfromsense，isverymuchstrainedandpuzzledtoframeclearideasoftheleastparticlesoftime，ortheleastincrementsgeneratedtherein：andmuchmoresotocomprehendthemoments，orthoseincrementsoftheflowingquantitiesinstatunascenti，intheirveryfirstoriginorbeginningtoexist，beforetheybecomefiniteparticles。Anditseemsstillmoredifficulttoconceivetheabstractedvelocitiesofsuchnascentimperfectentities。
Butthevelocitiesofthevelocities，thesecond，third，fourth，andfifthvelocities，&；c。，exceed，ifImistakenot，allhumanunderstanding。
Thefurtherthemindanalysethandpursueththesefugitiveideasthemoreitislostandbewildered；theobjects，atfirstfleetingandminute，soonvanishingoutofsight。Certainly，inanysense，asecondorthirdfluxionseemsanobscureMystery。Theincipientcelerityofanincipientcelerity，thenascentaugmentofanascentaugment，i。e。ofathingwhichhathnomagnitude：takeitinwhatlightyouplease，theclearconceptionofitwill，ifImistakenot，befoundimpossible；whetheritbesoornoIappealtothetrialofeverythinkingreader。Andifasecondfluxionbeinconceivable，whatarewetothinkofthird，fourth，fifthfluxions，andsoonwithoutend？
5。Theforeignmathematiciansaresupposedbysome，evenofourown，toproceedinamannerlessaccurate，perhaps，andgeometrical，yetmoreintelligible。Insteadofflowingquantitiesandtheirfluxions，theyconsiderthevariablefinitequantitiesasincreasingordiminishingbythecontinualadditionorsubductionofinfinitelysmallquantities。
Insteadofthevelocitieswherewithincrementsaregenerated，theyconsidertheincrementsordecrementsthemselves，whichtheycalldifferences，andwhicharesupposedtobeinfinitelysmall。Thedifferenceofalineisaninfinitelylittleline：ofaplaneaninfinitelylittleplane。Theysupposefinitequantitiestoconsistofpartsinfinitelylittle，andcurvestobepolygons，whereofthesidesareinfinitelylittle，whichbytheanglestheymakeonewithanotherdeterminethecurvityoftheline。Nowtoconceiveaquantityinfinitelysmall，thatis，infinitelylessthananysensibleorimaginablequantity，oranytheleastfinitemagnitude，is，Iconfess，abovemycapacity。Buttoconceiveapartofsuchinfinitelysmallquantitythatshallbestillinfinitelylessthanit，andconsequentlythoughmultipliedinfinitelyshallneverequaltheminutestfinitequantity，is，Isuspect，aninfinitedifficultytoanymanwhatsoever；andwillbeallowedsuchbythosewhocandidlysaywhattheythink；providedtheyreallythinkandreflect，anddonottakethingsupontrust。
6。Andyetinthecalculusdifferentialis，whichmethodservestoallthesameintentsandendswiththatoffluxions，ourmodernanalystsarenotcontenttoconsideronlythedifferencesoffinitequantities：theyalsoconsiderthedifferencesofthosedifferences，andthedifferencesofthedifferencesofthefirstdifferences：andsoonadinfinitum。Thatis，theyconsiderquantitiesinfinitelylessthantheleastdiscerniblequantity；andothersinfinitelylessthanthoseinfinitelysmallones；andstillothersinfinitelylessthantheprecedinginfinitesimals，andsoonwithoutendorlimit。Insomuchthatwearetoadmitaninfinitesuccessionofinfinitesimals，eachinfinitelylessthantheforegoing，andinfinitelygreaterthanthefollowing。Astherearefirst，second，third，fourth，fifth&；c。fluxions，sotherearedifferences，first，second，thirdfourth，&；c。inaninfiniteprogressiontowardsnothing，whichyoustillapproachandneverarriveat。And（whichismoststrange）althoughyoushouldtakeamillionofmillionsoftheseinfinitesimals，eachwhereofissupposedinfinitelygreaterthansomeotherrealmagnitude，andaddthemtotheleastgivenquantity，itshallneverbethebigger。
Forthisisoneofthemodestpostulataofourmodernmathematicians，andisacorner-stoneorground-workoftheirspeculations。
7。Allthesepoints，Isay，aresupposedandbelievedbycertainrigorousexactorsofevidenceinreligion，menwhopretendtobelievenofurtherthantheycansee。Thatmenwhohavebeenconversantonlyaboutclearpointsshouldwithdifficultyadmitobscureonesmightnotseemaltogetherunaccountable。Buthewhocandigestasecondorthirdfluxion，asecondorthirddifference，neednot，methinks，besqueamishaboutanypointinDivinity。Thereisanaturalpresumptionthatmen’sfacultiesaremadealike。Itisonthissuppositionthattheyattempttoargueandconvinceoneanother。Whatthereforeshallappearevidentlyimpossibleandrepugnanttoonemaybepresumedthesametoanother。Butwithwhatappearanceofreasonshallanymanpresumetosaythatmysteriesmaynotbeobjectsoffaith，atthesametimethathehimselfadmitssuchobscuremysteriestobetheobjectofscience？
8。Itmustindeedbeacknowledgedthemodernmathematiciansdonotconsiderthesepointsasmysteries，butasclearlyconceivedandmasteredbytheircomprehensiveminds。Theyscruplenottosaythatbythehelpofthesenewanalyticstheycanpenetrateintoinfinityitself：
thattheycanevenextendtheirviewsbeyondinfinity：thattheirartcomprehendsnotonlyinfinite，butinfiniteofinfinite（astheyexpressit），oraninfinityofinfinites。But，notwithstandingalltheseassertionsandpretensions，itmaybejustlyquestionedwhether，asothermeninotherinquiriesareoftendeceivedbywordsorterms，sotheylikewisearenotwonderfullydeceivedanddeludedbytheirownpeculiarsigns，symbolsorspecies。Nothingiseasierthantodeviseexpressionsornotationsforfluxionsandinfinitesimalsofthefirst，second，third，fourth，andsubsequentorders，proceedinginthesameregularformwithoutendorlimit，，，&；c。，ordx，ddx，dddx，ddddx&；c。Theseexpressions，indeed，areclearanddistinct，andthemindfindsnodifficultyinconceivingthemtobecontinuedbeyondanyassignablebounds。Butifweremovetheveilandlookunderneath，if，layingasidetheexpressions，wesetourselvesattentivelytoconsiderthethingsthemselveswhicharesupposedtobeexpressedormarkedthereby，weshalldiscovermuchemptiness，darkness，andconfusion；nay，ifImistakenot，directimpossibilitiesandcontradictions。Whetherthisbethecaseorno，everythinkingreaderisentreatedtoexamineandjudgeforhimself。
9。Havingconsideredtheobject，Iproceedtoconsidertheprinciplesofthisnewanalysisbymomentums，fluxions，orinfinitesimals；
whereinifitshallappearthatyourcapitalpoints，uponwhichtherestaresupposedtodepend，includeerrorandfalsereasoning；itwillthenfollowthatyou，whoareatalosstoconductyourselves，cannotwithanydecencysetupforguidestoothermen。Themainpointinthemethodoffluxionsistoobtainthefluxionormomentumoftherectangleorproductoftwoindeterminatequantities。Inasmuchasfromthencearederivedrulesforobtainingthefluxionsofallotherproductsandpowers；bethecoefficientsortheindexeswhattheywill，integersorfractions，rationalorsurd。
Now，thisfundamentalpointonewouldthinkshouldbeveryclearlymadeout，consideringhowmuchisbuiltuponit，andthatitsinfluenceextendsthroughoutthewholeanalysis。Butletthereaderjudge。Thisisgivenfordemonstration。[`NaturalisPhilosophiaePrincipiaMathematica，’lib。
ii。，lem。2。]SupposetheproductorrectangleABincreasedbycontinualmotion：andthatthemomentaneousincrementsofthesidesAandBareaandb。WhenthesidesAandBweredeficient，orlesserbyonehalfoftheirmoments，therectanglewas，i。e。，。AndassoonasthesidesAandBareincreasedbytheothertwohalvesoftheirmoments，therectanglebecomesor。Fromthelatterrectanglesubducttheformer，andtheremainingdifferencewillbeaB+bA。ThereforetheincrementoftherectanglegeneratedbytheentireincrementsaandbisaB+bA。Q。E。D。ButitisplainthatthedirectandtruemethodtoobtainthemomentorincrementoftherectangleAB，istotakethesidesasincreasedbytheirwholeincrements，andsomultiplythemtogether，A+abyB+b，theproductwhereofAB+aB+bA+abistheaugmentedrectangle；
whence，ifwesubductABtheremainderaB+bA+abwillbethetrueincrementoftherectangle，exceedingthatwhichwasobtainedbytheformerillegitimateandindirectmethodbythequantityab。
Andthisholdsuniversallybethequantitiesaandbwhattheywill，bigorlittle，finiteorinfinitesimal，increments，moments，orvelocities。Norwillitavailtosaythatabisaquantityexceedingsmall：sincewearetoldthatinrebusmathematiciserroresquamminiminonsuntcontemnendi。
10。Such[`Introd。adQuadraturamCurvarum。’]reasoningasthisfordemonstration，nothingbuttheobscurityofthesubjectcouldhaveencouragedorinducedthegreatauthorofthefluxionarymethodtoputuponhisfollowers，andnothingbutanimplicitdeferencetoauthoritycouldmovethemtoadmit。Thecaseindeedisdifficult。Therecanbenothingdonetillyouhavegotridofthequantityab。Inordertothisthenotionoffluxionsisshifted：itisplacedinvariouslights：pointswhichshouldbeasclearasfirstprinciplesarepuzzled；andtermswhichshouldbesteadilyusedareambiguous。But，notwithstandingallthisaddressandskill，thepointofgettingridofabcannotbeobtainedbylegitimatereasoning。Ifaman，bymethodsnotgeometricalordemonstrative，shallhavesatisfiedhimselfoftheusefulnessofcertainrules；whichheafterwardsshallproposetohisdisciplesforundoubtedtruths；whichheundertakestodemonstrateinasubtilemanner，andbythehelpofniceandintricatenotions；itisnothardtoconceivethatsuchhisdisciplesmay，tosavethemselvesthetroubleofthinking，beinclinedtoconfoundtheusefulnessofarulewiththecertaintyofatruth，andaccepttheonefortheother；especiallyiftheyaremenaccustomedrathertocomputethantothink；earnestrathertogoonfastandfar，thansolicitoustosetoutwarilyandseetheirwaydistinctly。
11。Thepointsormerelimitsofnascentlinesareundoubtedlyequal，ashavingnomoremagnitudeonethananother，alimitassuchbeingnoquantity。Ifbyamomentumyoumeanmorethantheveryinitiallimit，itmustbeeitherafinitequantityoraninfinitesimal。
Butallfinitequantitiesareexpresslyexcludedfromthenotionofamomentum。
Thereforethemomentummustbeaninfinitesimal。And，indeed，thoughmuchartificehathbeenemployedtoescapeoravoidtheadmissionofquantitiesinfinitelysmall，yetitseemsineffectual。ForaughtIsee，youcanadmitnoquantityasamediumbetweenafinitequantityandnothing，withoutadmittinginfinitesimals。Anincrementgeneratedinafiniteparticleoftimeisitselfafiniteparticle；andcannotthereforebeamomentum。Youmustthereforetakeaninfinitesimalpartoftimewhereintogenerateyourmomentum。Itissaid，themagnitudeofmomentsisnotconsidered；andyetthesesamemomentsaresupposedtobedividedintoparts。Thisisnoteasytoconceive，nomorethanitiswhyweshouldtakequantitieslessthanAandBinordertoobtaintheincrementofAB，ofwhichproceedingitmustbeownedthefinalcauseormotiveisobvious；
butitisnotsoobviousoreasytoexplainajustandlegitimatereasonforit，orshowittobegeometrical。
12。Fromtheforegoingprinciple，sodemonstrated，thegeneralruleforfindingthefluxionofanypowerofaflowingquantityisderived。[`PhilosophiaeNaturalisPrincipiaMathematica，’lib。ii。，lem。2。]But，asthereseemstohavebeensomeinwardscrupleorconsciousnessofdefectintheforegoingdemonstration，andasthisfindingthefluxionofagivenpowerisapointofprimaryimportance，ithaththereforebeenjudgedpropertodemonstratethesameinadifferentmanner，independentoftheforegoingdemonstration。Butwhetherthisothermethodbemorelegitimateandconclusivethantheformer，Iproceednowtoexamine；andinordertheretoshallpremisethefollowinglemma：-`If，withaviewtodemonstrateanyproposition，acertainpointissupposed，byvirtueofwhichcertainotherpointsareattained；andsuchsupposedpointbeitselfafterwardsdestroyedorrejectedbyacontrarysupposition；inthatcase，alltheotherpointsattainedthereby，andconsequentthereupon，mustalsobedestroyedandrejected，soasfromthenceforwardtobenomoresupposedorappliedinthedemonstration。’Thisissoplainastoneednoproof。
13。Now，theothermethodofobtainingaruletofindthefluxionofanypowerisasfollows。Letthequantityxflowuniformly，andbeitproposedtofindthefluxionof。
Inthesametimethatxbyflowingbecomesx+o，thepowerbecomes，i。e。bythemethodofinfiniteseriesandtheincrementsaretooneanotherasLetnowtheincrementsvanish，andtheirlastproportionwillbe1to。
Butitshouldseemthatthisreasoningisnotfairorconclusive。Forwhenitissaid，lettheincrementsvanish，i。e。lettheincrementsbenothing，orlettherebenoincrements，theformersuppositionthattheincrementsweresomething，orthattherewereincrements，isdestroyed，andyetaconsequenceofthatsupposition，i。e。anexpressiongotbyvirtuethereof，isretained。Which，bytheforegoinglemma，isafalsewayofreasoning。Certainlywhenwesupposetheincrementstovanish，wemustsupposetheirproportions，theirexpressions，andeverythingelsederivedfromthesuppositionoftheirexistencetovanishwiththem。
14。Tomakethispointplainer，Ishallunfoldthereasoning，andproposeitinafullerlighttoyourview。Itamountsthereforetothis，ormayinotherwordsbethusexpressed。Isupposethatthequantityxflows，andbyflowingisincreased，anditsincrementIcallo，sothatbyflowingitbecomesx+o。Andasxincreaseth，itfollowsthateverypowerofxislikewiseincreasedinadueproportion。Thereforeasxbecomesx+o，willbecome，thatis，accordingtothemethodofinfiniteseries，Andiffromthetwoaugmentedquantitieswesubducttherootandthepowerrespectively，weshallhaveremainingthetwoincrements，towit，whichincrements，beingbothdividedbythecommondivisoro，yieldthequotientswhicharethereforeexponentsoftheratiooftheincrements。HithertoIhavesupposedthatxflows，thatxhatharealincrement，thatoissomething。AndIhaveproceededallalongonthatsupposition，withoutwhichIshouldnothavebeenabletohavemadesomuchasonesinglestep。FromthatsuppositionitisthatIgetattheincrementof，thatIamabletocompareitwiththeincrementofx，andthatI
findtheproportionbetweenthetwoincrements。Inowbegleavetomakeanewsuppositioncontrarytothefirst，i。e。Iwillsupposethatthereisnoincrementofx，orthatoisnothing；whichsecondsuppositiondestroysmyfirst，andisinconsistentwithit，andthereforewitheverythingthatsupposethit。Idoneverthelessbegleavetoretain，whichisanexpressionobtainedinvirtueofmyfirstsupposition，whichnecessarilypresupposedsuchsupposition，andwhichcouldnotbeobtainedwithoutit。Allwhichseemsamostinconsistentwayofarguing，andsuchaswouldnotbeallowedofinDivinity。
15。Nothingisplainerthanthatnojustconclusioncanbedirectlydrawnfromtwoinconsistentsuppositions。Youmayindeedsupposeanythingpossible；butafterwardsyoumaynotsupposeanythingthatdestroyswhatyoufirstsupposed：or，ifyoudo，youmustbegindenovo。Ifthereforeyousupposethattheaugmentsvanish，i。e。thattherearenoaugments，youaretobeginagainandseewhatfollowsfromsuchsupposition。Butnothingwillfollowtoyourpurpose。Youcannotbythatmeanseverarriveatyourconclusion，orsucceedinwhatiscalledbythecelebratedauthor，theinvestigationofthefirstorlastproportionsofnascentandevanescentquantities，byinstitutingtheanalysisinfiniteones。Irepeatitagain：youareatlibertytomakeanypossiblesupposition：
andyoumaydestroyonesuppositionbyanother：butthenyoumaynotretaintheconsequences，oranypartoftheconsequences，ofyourfirstsuppositionsodestroyed。Iadmitthatsignsmaybemadetodenoteeitheranythingornothing：andconsequentlythatintheoriginalnotationx+o，omighthavesignifiedeitheranincrementornothing。Butthen，whichofthesesoeveryoumakeitsignify，youmustargueconsistentlywithsuchitssignification，andnotproceeduponadoublemeaning：whichtodowereamanifestsophism。Whetheryouargueinsymbolsorinwordstherulesofrightreasonarestillthesame。Norcanitbesupposedyouwillpleadaprivilegeinmathematicstobeexemptfromthem。
16。Ifyouassumeatfirstaquantityincreasedbynothing，andintheexpressionx+o，ostandsfornothing，uponthissupposition，asthereisnoincrementoftheroot，sotherewillbenoincrementofthepower；andconsequentlytherewillbenoneexceptthefirstofallthosemembersoftheseriesconstitutingthepowerofthebinomial；youwillthereforenevercomeatyourexpressionofafluxionlegitimatelybysuchmethod。Henceyouaredrivenintothefallaciouswayofproceedingtoacertainpointonthesuppositionofanincrement，andthenatonceshiftingyoursuppositiontothatofnoincrement。
Theremayseemgreatskillindoingthisatacertainpointorperiod。
Since，ifthissecondsuppositionhadbeenmadebeforethecommondivisionbyo，allhadvanishedatonce，andyoumusthavegotnothingbyyoursupposition。Whereas，bythisartificeoffirstdividingandthenchangingyoursupposition，youretain1and。
But，notwithstandingallthisaddresstocoverit，thefallacyisstillthesame。For，whetheritbedonesoonerorlater，whenoncethesecondsuppositionorassumptionismade，inthesameinstanttheformerassumptionandallthatyougotbyitisdestroyed，andgoesouttogether。Andthisisuniversallytrue，bethesubjectwhatitwill，throughoutallthebranchesofhumanknowledge；inanyotherofwhich，Ibelieve，menwouldhardlyadmitsuchareasoningasthis，whichinmathematicsisacceptedfordemonstration。
17。Itmaynotbeamisstoobservethatthemethodforfindingthefluxionofarectangleoftwoflowingquantities，asitissetforthintheTreatiseofQuadratures，differsfromtheabove-mentionedtakenfromthesecondbookofthePrinciples，andisineffectthesamewiththatusedinthecalculusdifferentialis。[`AnalysedesInfinimentPetits，’PartI。，prop。2。]Forthesupposingaquantityinfinitelydiminished，andthereforerejectingit，isineffecttherejectinganinfinitesimal；
andindeeditrequiresamarvelloussharpnessofdiscernmenttobeabletodistinguishbetweenevanescentincrementsandinfinitesimaldifferences。
Itmayperhapsbesaidthatthequantitybeinginfinitelydiminishedbecomesnothing，andsonothingisrejected。But，accordingtothereceivedprinciples，itisevidentthatnogeometricalquantitycanbyanydivisionorsubdivisionwhatsoeverbeexhausted，orreducedtonothing。Consideringthevariousartsanddevicesusedbythegreatauthorofthefluxionarymethod；inhowmanylightsheplacethhisfluxions；andinwhatdifferentwaysheattemptstodemonstratethesamepoint；onewouldbeinclinedtothink，hewashimselfsuspiciousofthejustnessofhisowndemonstrations，andthathewasnotenoughpleasedwithanynotionsteadilytoadheretoit。
Thusmuchatleastisplain，thatheownedhimselfsatisfiedconcerningcertainpointswhichneverthelesshewouldnotundertaketodemonstratetoothers。[See`LettertoJohnCollins，’Nov。8，1676。]Whetherthissatisfactionarosefromtentativemethodsorinductions，whichhaveoftenbeenadmittedbymathematicians（forinstance，byDr。Wallis，inhisArithmeticofInfinites），iswhatIshallnotpretendtodetermine。But，whateverthecasemighthavebeenwithrespecttotheauthor，itappearsthathisfollowershaveshownthemselvesmoreeagerinapplyinghismethod，thanaccurateinexamininghisprinciples。
18。Itiscurioustoobservewhatsubtletyandskillthisgreatgeniusemploystostrugglewithaninsuperabledifficulty；
andthroughwhatlabyrinthsheendeavourstoescapethedoctrineofinfinitesimals；
whichasitintrudesuponhimwhetherhewillorno，soitisadmittedandembracedbyotherswithouttheleastrepugnance；Leibnitzandhisfollowersintheircalculusdifferentialismakingnomannerofscruple，firsttosuppose，andsecondlytoreject，quantitiesinfinitelysmall；withwhatclearnessintheapprehensionandjustnessinthereasoning，anythinkingman，whoisnotprejudicedinfavourofthosethings，mayeasilydiscern。
Thenotionorideaofaninfinitesimalquantity，asitisanobjectsimplyapprehendedbythemind，hathalreadybeenconsidered。[Sect。5
and6。]Ishallnowonlyobserveastothemethodofgettingridofsuchquantities，thatitisdonewithouttheleastceremony。Asinfluxionsthepointoffirstimportance，andwhichpavesthewaytotherest，istofindthefluxionofaproductoftwoindeterminatequantities，sointhecalculusdifferentialis（whichmethodissupposedtohavebeenborrowedfromtheformerwithsomesmallalterations）themainpointistoobtainthedifferenceofsuchproduct。Nowtheruleforthisisgotbyrejectingtheproductorrectangleofthedifferences。Andingeneralitissupposedthatnoquantityisbiggerorlesserfortheadditionorsubductionofitsinfinitesimal：andthatconsequentlynoerrorcanarisefromsuchrejectionofinfinitesimals。
19。Andyetitshouldseemthat，whatevererrorsareadmittedinthepremises，proportionalerrorsoughttobeapprehendedintheconclusion，betheyfiniteorinfinitesimal：andthatthereforetheofgeometryrequiresnothingshouldbeneglectedorrejected。Inanswertothisyouwillperhapssay，thattheconclusionsareaccuratelytrue，andthatthereforetheprinciplesandmethodsfromwhencetheyarederivedmustbesotoo。Butthisinvertedwayofdemonstratingyourprinciplesbyyourconclusions，asitwouldbepeculiartoyougentlemen，soitiscontrarytotherulesoflogic。Thetruthoftheconclusionwillnotproveeithertheformorthematterofasyllogismtobetrue；inasmuchastheillationmighthavebeenwrongorthepremisesfalse，andtheconclusionneverthelesstrue，thoughnotinvirtueofsuchillationorofsuchpremises。
Isaythatineveryothersciencemenprovetheirconclusionsbytheirprinciples，andnottheirprinciplesbytheconclusions。Butifinyoursyoushouldallowyourselvesthisunnaturalwayofproceeding，theconsequencewouldbethatyoumusttakeupwithInduction，andbidadieutoDemonstration。
Andifyousubmittothis，yourauthoritywillnolongerleadthewayinpointsofReasonandScience。
20。Ihavenocontroversyaboutyourconclusions，butonlyaboutyourlogicandmethod：howyoudemonstrate？whatobjectsyouareconversantwith，andwhetheryouconceivethemclearly？whatprinciplesyouproceedupon；howsoundtheymaybe；andhowyouapplythem？ItmustberememberedthatIamnotconcernedaboutthetruthofyourtheorems，butonlyaboutthewayofcomingatthem；whetheritbelegitimateorillegitimate，clearorobscure，scientificortentative。Topreventallpossibilityofyourmistakingme，Ibegleavetorepeatandinsist，thatIconsiderthegeometricalanalystasalogician，i。e。sofarforthashereasonsandargues；andhismathematicalconclusions，notinthemselves，butintheirpremises；notastrueorfalse，usefulorinsignificant，butasderivedfromsuchprinciples，andbysuchinferences。And，forasmuchasitmayperhapsseemanunaccountableparadoxthatmathematiciansshoulddeducetruepropositionsfromfalseprinciples，berightintheconclusionandyeterrinthepremises；Ishallendeavourparticularlytoexplainwhythismaycometopass，andshowhowerrormaybringforthtruth，thoughitcannotbringforthscience。
21。Inorderthereforetoclearupthispoint，wewillsupposeforinstancethatatangentistobedrawntoaparabola，andexaminetheprogressofthisaffairasitisperformedbyinfinitesimaldifferences。LetABbeacurve，theabscissaAP=x，theordinatePB=y，thedifferenceoftheabscissaPM=dx，thedifferenceoftheordinateRN=dy。Now，bysupposingthecurvetobeapolygon，andconsequentlyBN，theincrementordifferenceofthecurvetobeastraightlinecoincidentwiththetangent，andthedifferentialtriangleBRNtobesimilartothetriangleTPB，thesubtangentPTisfoundafourthproportionaltoRN：RB：PB：thatis，tody：dx：y。HencethesubtangentwillbeButhereinthereisanerrorarisingfromtheaforementionedfalsesupposition，whencethevalueofPTcomesoutgreaterthanthetruth：forinrealityitisnotthetriangleRNBbutRLBwhichissimilartoPBT，andtherefore（insteadofRN）RLshouldhavebeenthefirsttermoftheproportion，i。e。RN+NL，i。e。dy+z：whencethetrueexpressionforthesubtangentshouldhavebeenTherewasthereforeanerrorofdefectinmakingdythedivisor；
whicherrorwasequaltoz，i。e。NLthelinecomprehendedbetweenthecurveandthetangent。Nowbythenatureofthecurveyy=px，supposingptobetheparameter，whencebytheruleofdifferences2ydy=pdxandButifyoumultiplyy+dybyitself，andretainthewholeproductwithoutrejectingthesquareofthedifference，itwillthencomeout，bysubstitutingtheaugmentedquantitiesintheequationofthecurve，thattruly。Therewasthereforeanerrorofexcessinmakingwhichfollowedfromtheerroneousruleofdifferences。AndthemeasureofthisseconderrorisThereforethetwoerrorsbeingequalandcontrarydestroyeachother；thefirsterrorofdefectbeingcorrectedbyaseconderrorofexcess。
22。Ifyouhadcommittedonlyoneerror，youwouldnothavecomeatatruesolutionoftheproblem。Butbyvirtueofatwofoldmistakeyouarrive，thoughnotatscience，yetattruth。Forscienceitcannotbecalled，whenyouproceedblindfold，andarriveatthetruthnotknowinghoworbywhatmeans。TodemonstratethatzisequaltoletBRordxbemandRNordyben。
Bythethirty-thirdpropositionofthefirstbookoftheConicsofApollonius，andfromsimilartriangles，as2xtoysoismtoLikewisefromthenatureoftheparabolayy+2yn+nn=xp+mp，and2yn+nn=mp：whereforeandbecauseyy=px，willbeequaltox。Thereforesubstitutingthesevaluesinsteadofmandxweshallhavei。e。whichbeingreducedgives23。Now，Iobserve，inthefirstplace，thattheconclusioncomesoutright，notbecausetherejectedsquareofdywasinfinitelysmall，butbecausethiserrorwascompensatedbyanothercontraryandequalerror。Iobserve，inthesecondplace，thatwhateverisrejected，beiteverysosmall，ifitbereal，andconsequentlymakesarealerrorinthepremises，itwillproduceaproportionalrealerrorintheconclusion。
Yourtheoremsthereforecannotbeaccuratelytrue，noryourproblemsaccuratelysolved，invirtueofpremiseswhichthemselvesarenotaccurate；itbeingaruleinlogicthatconclusiosequiturpartemdebiliorem。Therefore，Iobserve，inthethirdplace，thatwhentheconclusionisevidentandthepremisesobscure，ortheconclusionaccurateandthepremisesinaccurate，wemaysafelypronouncethatsuchconclusionisneitherevidentnoraccurate，invirtueofthoseobscureinaccuratepremisesorprinciples；butinvirtueofsomeotherprinciples，whichperhapsthedemonstratorhimselfneverkneworthoughtof。Iobserve，inthelastplace，thatincasethedifferencesaresupposedfinitequantitieseversogreat，theconclusionwillneverthelesscomeoutthesame：inasmuchastherejectedquantitiesarelegitimatelythrownout，notfortheirsmallness，butforanotherreason，towit，becauseofcontraryerrors，which，destroyingeachother，do，uponthewhole，causethatnothingisreally，thoughsomethingis，apparently，thrownout。Andthisreasonholdsequallywithrespecttoquantitiesfiniteaswellasinfinitesimal，greataswellassmall，afootorayardlongaswellastheminutestincrement。
24。Forthefullerillustrationofthispoint，Ishallconsideritinanotherlight，andproceedinginfinitequantitiestotheconclusion，Ishallonlythenmakeuseofoneinfinitesimal。SupposethestraightlineMQcutsthecurveATinthepointsRandS。SupposeLRatangentatthepointR，ANtheabscissa，NRandOSordinates。LetANbeproducedtoO，andRPbedrawnparalleltoNO。
SupposeAN=x，NR=y，NO=v，PS=z，thesubsecantMN=s。Lettheequationy=xxexpressthenatureofthecurve：andsupposingyandxincreasedbytheirfiniteincrementswegety+z=xx+2xv+vv；whencetheformerequationbeingsubducted，thereremainsz=2xv+vv。Andbyreasonofsimilartriangleswhereinifforyandzwesubstitutetheirvalues，wegetAndsupposingNOtobeinfinitelydiminished，thesubsecantNMwillinthatcasecoincidewiththesubtangentNL，andvasaninfinitesimalmayberejected，whenceitfollowsthatwhichisthetruevalueofthesubtangent。And，sincethiswasobtainedbyoneonlyerror，i。e。byonceejectingoneonlyinfinitesimal，itshouldseem，contrarytowhathathbeensaid，thataninfinitesimalquantityordifferencemaybeneglectedorthrownaway，andtheconclusionneverthelessbeaccuratelytrue，althoughtherewasnodoublemistakeorrectifyingofoneerrorbyanother，asinthefirstcase。But，ifthispointbethoroughlyconsidered，weshallfindthereisevenhereadoublemistake，andthatonecompensatesorrectifiestheother。For，inthefirstplace，itwassupposedthatwhenNOisinfinitelydiminishedorbecomesaninfinitesimalthenthesubsecantNMbecomesequaltothesubtangentNL。Butthisisaplainmistake；foritisevidentthatasasecantcannotbeatangent，soasubsecantcannotbeasubtangent。
Bethedifferenceeversosmall，yetstillthereisadifference。And，ifNObeinfinitelysmall，therewilleventhenbeaninfinitelysmalldifferencebetweenNMandNL。ThereforeNMorswastoolittleforyoursupposition（whenyousupposeditequaltoNL）；andthiserrorwascompensatedbyaseconderrorinthrowingoutv，whichlasterrormadesbiggerthanitstruevalue，andinlieuthereofgavethevalueofthesubtangent。Thisisthetruestateofthecase，howeveritmaybedisguised。Andtothisinrealityitamounts，andisatbottomthesamething，ifweshouldpretendtofindthesubtangentbyhavingfirstfound，fromtheequationofthecurveandsimilartriangles，ageneralexpressionforallsubsecants，andthenreducingthesubtangentunderthisgeneralrule，byconsideringitasthesubsecantwhenvvanishesorbecomesnothing。
25。UponthewholeIobserve，First，thatvcanneverbenothing，solongasthereisasecant。Secondly，thatthesamelinecannotbebothtangentandsecant。Thirdly，thatwhenvandNO[Seetheforegoingfigure]vanisheth，PSandSRdoalsovanish，andwiththemtheproportionalityofthesimilartriangles。
Consequentlythewholeexpression，whichwasobtainedbymeansthereofandgroundedthereupon，vanishethwhenvvanisheth。Fourthly，thatthemethodforfindingsecantsortheexpressionofsecants，beiteversogeneral，cannotincommonsenseextendanyfartherthantoallsecantswhatsoever：and，asitnecessarilysupposedsimilartriangles，itcannotbesupposedtotakeplacewheretherearenotsimilartriangles。Fifthly，thatthesubsecantwillalwaysbelessthanthesubtangent，andcannevercoincidewithit；whichcoincidencetosupposewouldbeabsurd；foritwouldbesupposingthesamelineatthesametimetocutandnottocutanothergivenline；whichisamanifestcontradiction，suchassubvertsthehypothesisandgivesademonstrationofitsfalsehood。Sixthly，ifthisbenotadmitted，Idemandareasonwhyanyotherapagogicaldemonstration，ordemonstrationadabsurdumshouldbeadmittedingeometryratherthanthis：orthatsomerealdifferencebeassignedbetweenthisandothersassuch。Seventhly，IobservethatitissophisticaltosupposeNOorRP，PS，andSRtobefinitereallinesinordertoformthetriangle，RPS，inordertoobtainproportionsbysimilartriangles；andafterwardstosupposetherearenosuchlines，norconsequentlysimilartriangles，andneverthelesstoretaintheconsequenceofthefirstsupposition，aftersuchsuppositionhathbeendestroyedbyacontraryone。
Eighthly，thatalthough，inthepresentcase，byinconsistentsuppositionstruthmaybeobtained，yetsuchtruthisnotdemonstrated：thatsuchmethodisnotconformabletotherulesoflogicandrightreason：that，howeverusefulitmaybe，itmustbeconsideredonlyasapresumption，asaknack，anart，ratheranartifice，butnotascientificdemonstration。
26。Thedoctrinepremisedmaybefurtherillustratedbythefollowingsimpleandeasycase，whereinIshallproceedbyevanescentincrements。SupposeAB=x，BC=y，BD=o，andthatxxisequaltotheareaABC：itisproposedtofindtheordinateyorBC。Whenxbyflowingbecomesx+o，thenxxbecomesxx+2xo+oo：
andtheareaABCbecomesADH，andtheincrementofxxwillbeequaltoBDHC，theincrementofthearea，i。e。toBCFD+CFH。AndifwesupposethecurvilinearspaceCFHtobeqoo，then2xo+oo=yo=qoo，whichdividedbyogive2x+o=y+qo。
And，supposingotovanish，2x=y，inwhichcaseACHwillbeastraightline，andtheareasABC，CFHtriangles。Nowwithregardtothisreasoning，ithathbeenalreadyremarked，[Sect。12and13supra。]thatitisnotlegitimateorlogicaltosupposeotovanish，i。e。tobenothing，i。e。thatthereisnoincrement，unlesswerejectatthesametimewiththeincrementitselfeveryconsequenceofsuchincrement，i。e。whatsoevercouldnotbeobtainedbysupposingsuchincrement。Itmustneverthelessbeacknowledgedthattheproblemisrightlysolved，andtheconclusiontrue，towhichweareledbythismethod。Itwillthereforebeasked，howcomesittopassthatthethrowingoutoisattendedwithnoerrorintheconclusion？
Ianswer，thetruereasonhereofisplainlythis：becauseqbeingunit，qoisequaltoo：andtherefore2x+o-qo=y=2x，theequalquantitiesqoandobeingdestroyedbycontrarysigns。