Andthisholdsuniversallybethequantitiesaandbwhattheywill，bigorlittle，finiteorinfinitesimal，increments，moments，orvelocities。Norwillitavailtosaythatabisaquantityexceedingsmall：Sincewearetoldthatinrebusmathematiciserroresquamminiminonsuntcontemnendi。[Introduct。adQuadrat。Curv。]Suchreasoningasthis，fordemonstration，nothingbuttheobscurityofthesubjectcouldhaveencouragedorinducedthegreatauthorofthefluxionarymethodtoputuponhisfollowers，andnothingbutanimplicitedeferencetoauthoritycouldmovethemtoadmit。Thecaseindeedisdifficult。Therecanbenothingdonetillyouhavegotridofthequantityab。InordertothisthenotionofFluxionsisshifted：itisplacedinvariouslights：PointswhichshouldbeasclearasfirstPrinciplesarepuzzled；andtermswhichshouldbesteadilyusedareambiguous。Butnotwithstandingallthisaddressandskillthepointofgettingridofabcannotbeobtainedbylegitimatereasoning。’’ItisnowtimetohearSirIsaacNewton。
Princip。Lib。IILemm。2。Cas。1。``RectangulumquodvismotuperpetuoauctumAB，ubidelateribusA&；Bdeerantmomentorumdimidia&；，fuitin，seu；&；quamprimumlateraA&；Balterismomentorumdimidiisauctasunt，evaditin，seu。Dehocrectangulosubducaturrectangulumprius，&；manebitexcessusaBbA。Igiturlaterumincrementistotisa&；bgeneraturrectanguliincrementumaBbA。Q。E。D。’’HavingnowfairlylaidbeforemyreaderwhatbothyourselfandSirIsaacNewtonhavedelivereduponthissubject，Icometoexaminewhichofyouisintheright。
Inthefirstplace，IfindyoutakeitforgrantedthatwhatSirIsaacNewtonishereendeavouringtofind，bysupposingthesidesAandBfirsttowanthalftheirmoments，andafterwardstohavegainedtheotherhalvesoftheirmoments，istheincrementoftherectangleAB。
InthisIconceiveyouaremistaken。Forneitherinthedemonstrationitself，norinanythingprecedingorfollowingit，isanymentionsomuchasoncemadeoftheincrementoftherectangleAB。Onthecontraryitplainlyappearsthatwhatheendeavourstoobtainbythesesuppositions，isnootherthantheincrementoftherectangle，andyoumustownhetakesthedirectandtruemethodtoobtainit。
Butyouwillsay，isitnotthebusinessofthislemmatodeterminethemomentsofflowingquantities？AndisnotthedesignofCase1todeterminethemomentoftherectangleAB？Ianswerthatisisso：butthatrigorouslyspeakingthemomentoftherectangleAB，isnot，asyousuppose，theincrementoftherectangleAB；butitistheincrementoftherectangle。Inordertoclearupthispoint，Imustobserve，ThatthewordmomentisusedbySirIsaacNewtonandyourselftosignifieindifferentlyeitheranincrement，oradecrement。ThataBbAabisbyyoudemonstratedtobethetrueincrementoftherectangleAB。ThataBbA-abisthetruedecrementofthesamerectangleAB；asplainlyappearsupontakingthesametrueanddirectmethodforfindingthedecrement，asyouhaveusedforfindingtheincrement。Now，Sir，Iwouldhumblybegleavetoinquireofyou，whoseesomuchmoreclearlyintothesemattersthanSirIsaacNewtonoranyofhisfollowers；
whichofthesetwoQuantitiesaBbAabandaBbA-ab，youwillbepleasedtocallthemomentoftherectangleAB？Thecaseindeedisdifficult，thedifferencebetweenthemisnolessthan2ab，justthedoubleofthesameab，whichhasgivenusallsomuchtrouble；andyeteachofthempleadanequalrighttothetitleofmoment。Soequalaone，that，thoughIamverysensibleofyouraddressandskill，yetthereseemstobenopossibilityofdecidingthecontroversybetweenthembylegitimatereasoning。
Iseebuttwowaysofdoingit。Oneisthattheyshouldtossupcrossorpileforthetitle：OrifthatbethoughttooboyishandunbeseemingtheGravityofMathematicalquantities，theymustevenendthedisputeinanamicablemanner，andwithoutclaiminganypreferenceoneofanother，agreethattheymaketwomomentsbetweenthem。Then，Sir，Iapprehendthecasewillstandthus：aBbAabaBbA-abmakingtwicethemomentoftherectangleAB；itfollowsthataBbAwillmakethesinglemomentofthesamerectangle。
Yousee，Sir，afterallthepainsyouhavetaken，thisaffaircomesout，evenuponyourownconcessions，justasSirIsaacNewtonandhisfollowerswouldhaveit。Believeme，thereisnoremedy。Youmustacquiesce。
OnlyifitmaybeanySatisfactiontoyoutoknowwhySirIsaactookthisindirectwayoffindingtheincrementof，insteadofproceedingdirectlytofindthemomentoftherectangleAB，Ishallbereadytoobligeyouasfarascanbeexpectedfromoneofthose，whohaveshownthemselvesmoreeagerinapplyinghismethod，thanaccurateinexamininghisprinciples。
Thefinalcauseormotivetothisproceeding，Ifind，isnotunknowntoyou；yousayitisveryobvious，meaning，Isuppose，thattherebyitwasintendedtoexcludethissametroublesomerectangleab。Whytruly，Sir，inabookofstrictdemonstration，asSirIsaacNewtonintendedhisPrincipiashouldbe，itwascertainlymorepropertoexcludethatquantity，soasnottosufferittoappear，thanfirsttointroduceitintothereader’sviewandthentorejectit。
Youaddthatitisnotsoobviousoreasytoexplainajustandlegitimatereasonforit，orshewittobeGeometrical。Howfaritmaybeobviousoreasytoassignsuchareason，Iwillnotdispute：thoughIamapttothinkthatwhatiseasytome，cannotbedifficulttootherpersons，providedtheyusethesameendeavourstofindthetruthasIhavedone。NowIapprehendthereasonofthisproceedingofSirIsaacNewtontobethefollowingveryplainone：ThatinordertofindthemomentoftherectangleAB，itismoreconsonanttostrictGeometricalrigourtotaketheincrementoftherectangle，thantotaketheincrementoftherectangleABitself。AndifI
canmakethisappear，youmustallowthathehadajustandlegitimatereasonforproceedingashedid。
YouknowverywellthatthemomentoftherectangleABisproportionaltothevelocityofthatrectangle，withwhichitalters，eitherinincreasing，orindiminishing。Now，Iask，inGeometricalrigour，whatisproperlythevelocityofthisrectangle？IsitthevelocitywithwhichtherectanglefromABbecomes；
orthevelocitywithwhichfromABitbecomes？
IfindmyselfexactlyinthecaseoftheAssbetweenthetwobottlesofhay：Iseenoreason，norpossibilityofareasontodeterminemeeitheroneway，ortheother。ButmethinksIhearthevenerableGhostofSirIsaacNewtonwhisperme，thatthevelocityIseekfor，isneithertheonenortheotherofthese，butisthevelocitywhichtheflowingrectanglehas，notwhileitisgreaterorlessthanAB，neitherbefore，norafteritbecomesAB，butattheveryinstantoftimethatitisAB。InlikemannerthemomentofthisrectangleisneithertheincrementfromABto；
norisitthedecrementfromABto：
ItisnotamomentcommontoABand，whichmaybeconsideredasantheincrementoftheformer，orasthedecrementofthelatter：NorisitamomentcommontoABand，whichmaybeconsideredasthedecrementofthefirst，orastheincrementofthelast：ButitisthemomentoftheveryindividualrectangleABitself，andpeculiartothatonly；andsuchasbeingconsideredindifferentlyeitherasanincrementordecrement，shallbeexactlyandperfectlythesame。AndthewaytoobtainsuchamomentisnottolookforonelyingbetweenABand；
nortolookforonelyingbetweenABand；
thatis，nottosupposeABaslyingateitherextremityofthemoment；
butasextendedtothemiddleofit；ashavingacquiredtheonehalfofthemoment，andasbeingabouttoacquiretheother；orashavinglostonehalfofit，andbeingabouttolosetheother。AndthisisthemethodSirIsaacNewtonhastakeninthedemonstrationyouexceptagainst。
Whatsayyou，Sir？IsthisajustandlegitimatereasonforSirIsaac’sproceedingashedid？Ithinkyoumustacknowledgeittobeso。ForevenifyoushouldstillhaveanydoubtwhetherhisproceedingberigorouslyGeometrical；yetyoucannotbutconfessthatwhethermomentsbeconsideredasinfinitelysmall，orasfinitequantities，hismethodapproachesnearertoGeometrickrigour，thanthatwhichyoupropose。I
thinklikewiseyoucannotbutbesensibleofgreatwantofcautioninyourownproceeding；inasmuchasthatquantity，whichSirIsaacNewtonthroughthiswholeLemma，andalltheseveralcasesofit，constantlycallsamoment，withoutconfiningittobeeitherincrementordecrement，isbyyouincosideratelyandarbitrarily，andwithoutanyshadowofreasongiven，supposedanddeterminedtobeanincrement。Andthis，Sir，naturallyleadsmetogiveyouapieceoffriendlyadvice，whichyouseemtostandmuchinneedof。Itisthat，wheneveryoutakeitintoyourheadtocriticiseuponSirIsaacNewton’swritings，youfirstexamineandweigheverywordheuses；andifyoutranslatehim，keepcloselytohisexpression。
Believeme，thisGreatMan，amonghisotherextraordinaryindowments，hadapeculiarsagacityinforeseeingobjections，aswellasanaversiontodisputing。Tothesetwoqualitiesaccompaniedwithextremehumanityandcondescensionitisowing，thatheusessuchaccuracyinhisexpression，thatanintelligentandattentivereadercannevermistakehim；andthathedoesofhimselffirstpropose，andthenremovesuchdifficulties，asmaynaturallyariseinthemindsofevencandidandjudiciouspersons，whoarenotyetmastersofthesubjecthetreatsof。ButasfortheHominesstolidi&；addepugnandumparati，hecontentshimselfwithobservingthatprudentcautionineverywordheuses，thatastheyshallfindnothingtomisleadthem，soontheotherhand，iftheyunreservedlyandunadvisedlyattackhim，theyshallcertainlyandunavoidablyinduereseinstimuloslatentes，andexposethemselvestothescornandcontemptofeveryunprejudicedobserver。
Thisgreatexample，whichinanythelowestdegreetoimitateisthehighesthonourIcaneverarriveat，orevendesire，movesmetoproposeandremoveanobjectionwhichmaypossiblyariseinyourmind，andhinderyoufromacquiescinginonepartofwhatIhavejustnowlaidbeforeyou。
ItisthatIhavesupposedtherectangleABextendedtothemiddleofitsmoment；ashavingacquiredthehalfofit，andbeingabouttoacquiretheother；orashavinglostonehalfofit，andbeingabouttolosetheother。Youmaysaythisisstrictlyandexactlytrueinrespectofthesidesofthatrectangle；whichsides，fromandarebecomeAandB；andareabouttobecomeand；
butthatitisnotequallytrueoftherectanglecomposedofthosesides，whichfrom，or，isbecomeAB；andisabouttobecome，or：sincethepartofthemomentwhichABissupposedtohavegained，namely，isnotequaltothatpartofthemomentwhichisabouttobegained，namely；thedifferencebetweenthembeing。
InanswertothisIreply，thatthesetwoquantities，andsolongasaandbarefinitequantities，areundoubtedlyunequal；butthatthemoreaandbarediminished，bysomuchnearerwillthesequantitiesapproachtoanequality；andifaandbarediminishedadinfinitum，thetwoquantitieswillthenbeperfectlyequal。SeethisdemonstratedPrincip。Lib。I。Sect。
I。Lemm。I。WhichLemma，foryourownsakeandmine，Icouldwishyouhadconsultedsooner。
Lastly，toremoveallscrupleanddifficultyaboutthisaffair，Imustobserve，thatthemomentoftherectangleAB，determinedbySirIsaacNewton，namelyaBbA，andtheincrementofthesamerectangle，determinedbyyourself，namelyaBbAab，areperfectlyandexactlyequal，supposingaandbtobediminishedadinfinitum；andthisbytheLemmajustnowquoted。
Icomenowtoyoursecondinstanceoffalsereasoning，whichyoutakefromtheBookofQuadratures；andpassingbytheLemmayousogravelylaydowntoshew，thatwhentwocontrarysuppositionsaremade，nothingcanbeinferredfromeitherofthem；asatruththatnoSchool-boycanbeignorantof；Ishallheretranscribethisinstanceoffalsereasoningasyougiveit，withyourobservationsuponit。``Letthequantityxflowuniformly，andbeitproposedtofindtheFluxionof。
Inthesametimethatxbyflowingbecomesxo，thepowerbecomes，i。e。bythemethodofinfiniteSeriesandtheincrementsoandaretooneanotheras1toLetnowtheincrementsvanish，andtheirlastproportionwillbe1to。
Butitshouldseemthatthisreasoningisnotfairorconclusive。Forwhenitissaid，lettheincrementsvanish，i。e。lettheincrementsbenothing，orlettherebenoincrements，theformersuppositionthattheincrementsweresomething，orthattherewereincrements，isdestroyed，andyetaconsequenceofthatsupposition，i。e。anexpressiongotbyvirtuethereof，ifretained。Which，bytheforegoingLemma，isafalsewayofarguing。Certainlywhenwesupposetheincrementstovanish，wemustsupposetheirproportions，theirexpressions，andeverythingelsederivedfromthesuppositionoftheirexistencetovanishwiththem。’’
[Analystp。20。]Youarepleasedtogoonforsomenumberofpages，tomakethispointplainer，tounfoldthereasoning，andtoproposeitinafullerlight。ButIthinkwemayaswellstophere。Youhavealreadysofullyunfoldedit，thatifthisbethewayofreasoningofourMathematicalInfidels，IpronounceourReligionoutofalldangerfromthatquarter。
FromthistimeourReverendClergymaysleepinquiet，andbeunderaslittleapprehensionfromtheunbelievingAnalyst，asfromthemostignorantofthePopishMonks，themoststupidoftheJewishRabbi’s，orthemostemptyandcontemptiblepratersamongtheMinutePhilosophers。Ihaveonlyonedoubtuponme。Pray，Sir，areyouverysurethatthisistherealdoctrineofSirIsaacNewton？Areyouabsolutelycertainyouhavenotmistakenhim？Youseem，Imustconfess，tobeexceedinglycautious，youblameothersfornotbeingaccurateinexamininghisPrinciples，youtalkofpreventingallpossibilityofmistakingyou，andyoutreathimandhisfollowersinsuchamanner，thatyouaretoexpectnoquarterfromthemincaseofillsuccess。Andyetthisissogreat，sounaccountable，sohorrid，sotrulyBoeotianablunder，thatIknownothowtothinkaGreatGenius，aNewtoncouldbeguiltyofit。ForGod’ssakeletusexamineitoncemore。Evanescantjamaugmentailla，letnowtheincrementsvanish，i。e。lettheincrementsbenothing，orlettherebenoincrements。Hold，Sir，Idoubtwearenotrighthere。IrememberSirIsaacNewtonoftenusesthetermsofmomentanascentiaandmomentaevanescentia。IthinkIhaveseenyoulikewiseseveraltimesusingtheliketermsofnascentandevanescentincrements。Also，ifIamnotmistaken，bothheandyouconsideranascent，orevanescentmoment，incrementordecrement，asthesamequantityunderdifferentcircumstances；
sometimesasinthepointofbeginningtoexist，andothertimesasinthepointofceasingtoexist。Fromthismethinksitshouldfollowthatthetwoexpressionssubjoined，willbeperfectlyequivalenttoeachother。Nascanturjamaugmentailla，&；eorumratioprimaeritEvanescentjamaugmentailla，&；eorumratioultimaeritThemeaningofthefirstcanpossiblybenootherthantoconsiderthefirstproportionbetweenthenascentaugments，inthepointoftheirbeginningtoexist。Mustnotthereforethemeaningofthelatterbetoconsiderthelastproportionbetweentheevanescentaugments，inthepointofevanescence，ortheirceasingtoexist？Oughtitnottobethustranslated，Lettheaugmentsnowbecomeevanescent，letthembeuponthepointofevanescence？
Whatthenmustwethinkofyourinterpretation，Lettheincrementsbenothing，lettherebenoincrements？Donotthewordsratioultimastareusintheface，andplainlytellusthatthoughthereisalastproportionofevanescentincrements，yettherecanbenoproportionofincrementswhicharenothing，ofincrementswhichdonotexist？Ibelieve，Sir，everythinkingpersonwillacquitSirIsaacNewtonofthegrossoversightyouascribetohim，andwillacknowledgethatitisyourselfalone，whohavebeenguiltyofamostpalpable，inexcusable，andunpardonableblunder。
Inowcometothethirdheadofyourobjections。
3。ArtsandfallaciesusedbySirIsaacNewtontomakehisfalsereasoningpassuponhisfollowers。
OnthisheadIshallnotneedtotakeupmuchofyourtime，becausehavingalreadyprovedthatSirIsaacNewtonwasnotguiltyoffalsereasoningintheinstancesyoualledge，Isupposenobodywillthinkhehadanyoccasiontomakeuseofartsandfallaciestoimposeuponhisfollowers。
Butyouhaveoneobservationuponthishead，whichissoverysingular，thatIcannotbutthinkitworthyofparticularconsideration。Considering，sayyou，thevariousartsanddevicesusedbytheGreatAuthoroftheFluxionarymethod：inhowmanylightsheplacethhisFluxions：andinwhatdifferentwaysheattemptstodemonstratethesamepoint：onewouldbeinclinedtothinkhewashimselfsuspiciousofthejustnessofhisowndemonstrations：andthathewasnotenoughpleasedwithanyonenotionsteadilytoadheretoit。Thusmuchatleastisplain，thatheownedhimselfsatisfiedconcerningcertainpoints，whichneverthelesshecouldnotundertaketodemonstratetoothers。[Analystp。27。]Really，Sir，thisseemstobeveryhardusage。SirIsaacNewtonhasmadeanewandgreatdiscovery，bywhichhehasnotonlyout-donealltheGeometriciansthateverwentbeforehim，butcanenablesuchordinaryproficientsinMathematicks，asyouandme，tosurpassallthegreatmastersofantiquity：Heissogoodastoinstructusinthismethod；andbecauseitrequiressomepainsanddiscernmenttocomprehenditrightly，hesetsitinseveralvariouslights，thatbymeansofsomeofthesewemaynotfailofunderstandingit。Pray，Sir，haveyouandIanyreasontocomplainofthis？Formypart，Ithinkmyselfgreatlyobligedtohimforhiscondescension：Ifhehadnottakensomuchpainstoexplainhisdoctrine，IdoubtIshouldneverhaveunderstoodit。But，forGod’ssake，whatisityouareoffendedat，whodonotstillunderstandhim？Youareallinthedark，andyetareangryathisgivingyousomuchlight。SurelythefaultisnotinSirIsaacNewton，butinyourowneyes。Sothickadropserenehasquench’dtheirorbs，Ordimsuffusionveil’d。Butisnothehimself，sayyou，suspiciousofthejustnessofhisowndemonstrations？
Pray，Sir，whenaDivineisinstructinghishearersinaweightyandimportantpointofReligion，iffromadesirethateveryoneshouldperfectlyunderstandhim，heisatpainstouseseveralarguments，andtosethisDoctrineinvariouslights；woulditbereasonable，orjust，orgratefulinanyofhisauditorstoinferfromthis，thatthePreacherwassuspiciousofthejustnessofhisownreasoning？Whenyou，afterallthedemonstrationsthathadbeengivenofthebeingofaGod，bythelearnedFathersoftheChurch，andbythewisestofthePhilosophersofallages，thoughtfittointroducethatnewandsingularoneofaVisualLanguage，woulditbefairinmetosupposethatyouweresuspiciousofalltheformerproofsoftheexistenceofaDeity，andleftthatgreatandimportanttruthtodependuponametaphoricalargument？Surelyoneargumentmaybejust，andconclusive，andperfectlysatisfactorytohimthatusesit；andyetthemattertreatedofmaybeofthatdifficulty，orofthatdignityandimportance，asnotonlytoadmitof，buttorequireseveralothersfortheinstructionandconvictionofhishearers。AndthusmuchmaysufficeforyourthirdandlastheadofobjectionsagainstSirIsaacNewtonandhisfollowers：OnlybeforeIconcludeImustadviseyoutocorrectonewordinyourextractfromhisLettertoMr。Collins，Nov。8，1676，[Analyst，p。27]orrathertogiveupthatextractintirely，asbeingofnomannerofservicetoyou。ThereisagreatdealofdifferencebetweensayingIcannotundertaketoproveathing，andIwillnotundertakeit。
SirIsaac，inthatLettersays，Iwillnot：Andbesides，thepointtherementionedisnotthepointhereindebate；sothatyouhavenorighttodrawanyinferencefromthatpointtothis。
HavingnowdonewitheverythingnecessarytothevindicationofSirIsaacNewtonandhisfollowers，andtherebydrivenyouentirelyoutofourintrenchments，IamconsideringwhetherIshouldsallyoutandattackyouinyourown。Youhavethrownupsomeworks，Isee，whichatadistancemakeaprettygoodappearance，andseemcapableofdefence：
Butupontakinganearerviewofthem，Ijudgethemtobeveryslightanduntenable，andtobeguardedratherbyanew-raised，undisciplinedMilitia，thananythingofveteran，regularTroops；sothatitwouldnotbeverydifficulttocarrythembyassault。Butastheyseemratherdesignedforshew，thanuse，moretoamuseyourself，thananywaytoannoyus，Iamdeterminedtoleaveyouinpossessionofthem。
OnlyyoursuppositionofadoubleerrorinthemethodofFluxions，[Analyst，p。31&；seq。top。49。]andtheuseyoumakeofittoshewhowtrueconclusionsareobtainedfromfalseprinciples，bymeansoftwocontrarymistakesexactlycompensatingoneanother，hassomethinginitsoextraordinary，astorequireanddeserveaparticularconsideration。ThisdarlingPhantom，thisbelovedoffspringofyourteemingbrain，whichlikeMinervaissuingarmedfromtheheadofJupiter，herspearinonehand，andherShieldwiththeGorgon’sheadintheother，istoturnallourMathematiciansintostocks，andstones，andstatues，issetforthwithsomuchartandskill，andisdressedoutinsoadvantageousandpompousamanner，todrawtheattentionandtodazzletheimaginationofthespectators，thatitwouldbeunpardonableneglectandrudenessinmetopassitbyunregarded。
Ishallnotthereforecontentmyselfwithsayingthatoneoftheseerrors[Seep。46-53。]isalreadybecomeevanescent，i。e。isnothing，isnoerroratall；andthattheotherofthemwilllikewiseimmediatelydisappearliketheGhostofadepartedquantity，[Analyst，p。59。]
ifyouexorciseitwithafewwordsoutofthefirstsectionofthePrincipia：
Onthecontrary，Iproposesofartogratifyyourfondnessforthishopefulscheme，astogiveitafairandfullexamination。
Wearetoconsiderthereforewhatmaybethereason，thatinthemethodofFluxionstheconclusionsareexactlytrue：Forintheexactnessoftheconclusionswearebothagreed；thoughtherebeawidedifferencebetweenusinrespectofthemeansbywhichMathematiciansarriveatthatexactness。
Iconceivethattheconclusionisthereforeexact，becauseitisdeducedbyjustreasoningfromcertainprinciples。YouonthecontraryareofopinionthatSirIsaacNewtonisguiltyofacapitalandfundamentalerroronrejectingthequantityab，sooftentalkedof，andthattheconclusioncomesoutright，notbecausethequantityrejectedisinfinitelysmall，butbecausethiserroriscompensatedbyanothercontraryandequalerror。[Analyst，p。35。]Andthisyousay，perhapstheDemonstratorhimselfneverkneworthoughtof。[p。36。]Ifhehadcommittedonlyoneerror，hewouldnothavecomeatatruesolutionoftheProblem。Butbyvirtueofatwo-foldmistakehearrives，thoughnotatscience，yetattruth。Forscienceitcannotbecalled，whenheproceedsblindfold，andarrivesatthetruthnotknowinghoworbywhatmeans。[p。34。]
Thisisthewayyouaccountforwhatyoujustlysay，mayperhapsseemanunaccountableParadox，thatMathematiciansshoulddeducetruePropositionsfromfalsePrinciples，berightintheconclusion，andyeterrinthepremisses；
thaterrorshouldbringforthtruth，thoughitcannotbringforthscience。
[p。31。]
Nowtruly，Sir，ifthisParadoxofyourshouldbewellmadeout，ImustconfessitoughtverymuchtoaltertheopiniontheworldhashadofSirIsaacNewton，andoccasionourtalkingofhiminaverydifferentmannerfromwhatwehavehithertodone。Whatthinkyouif，insteadofthegreatestthateverwas，weshouldcallhimthemostfortunate，themostluckyMathematicianthateverdrewacircle？MethinksIseethegoodoldGentlemanfastasleepandsnoringinhiseasychair，whileDameFortuneisbringinghimherapronfullofbeautifulTheoremsandProblemswhichheneverknowsorthinksof：justastheAtheniansoncepaintedherdraggingtownsandcitiestoherfavouriteGeneral。Forwhatelsebutextremegoodfortunecouldoccasiontheconclusionsarisingfromhismethodtobealwaystrueandjustandaccurate，whenthepremisseswereinaccurateanderroneousandfalse，andonlyledtorightconclusionsbymeansoftwoerrorsevercompensatingoneanothertotheutmostexactness？Whatluckwashere？Thatwhenhehadmadeonecapital，fundamental，generalmistake，heshouldhappentomakeasecond，ascapital，asfundamental，asgeneralasthefirst；Thatheshouldnotproceedtocommitthreeorfoursuchmistakes，butstopatthesecond：Thatthesetwomistakesshouldchancetolieboththesameway，butoncontrarysides，sothattheonemighthelptocorrecttheother；andlastly，thatthetwocontraryerrors，amongalltheinfiniteproportionswhichtheymightbeartooneanother，shouldhappenuponthatofaperfectequality；sothatonemightinallpossiblecasesbeexactlybalancedorcompensatedbytheother。Withaquarterofthisgoodfortuneamanmightgetthe10000l。prizeinthepresentLottery，withasingleTicket。
Buttocometoourpoint，wearetoexaminewhethertheexactnessoftheconclusionisowingtotheexactcompensationofoneoftheseerrorsbytheother，ortothoseerrorsbeingutterlyinsignificant，beinginrealitynoerrorsatall。AndinordertheretoIproposetoseehowtheconclusionwillcomeout，whenonlyoneoftheseerrorsiscommitted，sothatthereisnothingtocompensateit。
Inyour21section，whichwithitsfigureIherereferto，thefirsterrorissupposedtobethemakingthesubtangentorinsteadofTheseconderrorismakinginsteadofIfboththeseerrorsbecommitted，orifneitherofthembecommitted，theconclusionisagreedtobeequallyjustandright，givingS=2x。
IfIavoidthefirstoftheseerrors，bymakingandretainthesecond，bysupposingmyconclusionwillbeOntheotherhandifIcommitthefirsterror，andavoidthesecond，myconclusionwillgivemeNowIaffirmthatthesetwoseveralvaluesofS，whicharetheresultofoneerroronlywithoutanythingtocompensateit，arebothtrueandequallyexactwiththeformervalue，2x，whichistheresultofeitheroftwoerrors，orofnoneatall。You，Iamsensible，willdisputethiswithme，youwillsaythatoneofthese，islessthan2x；andthelatter，isinthesameproportionbiggerthan2x。Ibegleavetherefore，fortheinformationofsomeofmyreaders，toaskyouaquestion。Supposingthetruesubtangent2xtobeathousandmilesinlength，howmuchwillthesecondvalueofthatsubtangentfallshortofathousandmiles？Willitbeayard，orafoot，oraninch？
Noneoftheseyouconfess，northethousandth，northethousand-millionthpartofaninch。
Iaskfurther，whatthenisthisdifference？Isitpossibleinalltheinfinityoffractionalnumberstofindanythingsmallenoughtorepresentit？Youown，youconfessitisnot：Youmustconfesslikewise，thatifthesethreeseveralvaluesofSwerealltobeexpressedinnumbers，withoutbeingreducibletowhich，inyouropinion，[AnalystQuery24。]
theycanbeofnouse，theymusteveryonebeexpressedby1000，withouttheleasttittleofvariation，addition，ordiminution。Behold，GentleReader，whatamightybeam[MottotoAnalyst。]herehasbeendiscoveredintheeyesofMathematicians，incomparisonwithwhichallthedifficultiesinDivinityarebutasmotesandatoms！
Sincethereforetheseerrorsarewhollyinsignificant，myconclusionwhenreducedtonumbers，comingoutexactlythesame，whetherthefirst，orsecond，orneither，orbothoftheseerrorsbecommitted；andsincebycommittingboththeseerrors，thecalculus，whichwouldotherwise，especiallyinthehigheroperations，beexceedinglytediousandlaborious，isnowrenderedsurprisinglyexpeditiousandeasy；itseemstomethatthisissofarfrombeinganydefectinthemethodofFluxions，thatonthecontraryitisoneofthegreatestadvantagesandexcellenciesofthatinvention。
Butyoutellmeitisnottheusefulnessofthismethodthatisthematterindispute：allthequestioniswhetheritbescientifical，whetherthosewhouseit，seetheirwaydistinctly，orproceedblindfoldandarriveatthetruthnotknowinghoworbywhatmeans。Ihavespokentothisbefore，butmustaddawordortwomoreinthisplace。You，Sir，areforavoidingthesetwoerrors；Iamforretainingthem。Whenyouavoidthem，donotyouseeyourwaydistinctly？AndifIretainthem，voluntarily，andwithmyeyesopen；mayInotneverthelessclearlyseetheeffectoftheseerrors，orofeitherofthem，ineverystepItakeandintheconclusionIatlastcometo？MayInotthereforelikewisebesaidtoseemywaydistinctly？Now，ifyouandIcanseeourwaysowell，Iamafraiditwillbeconstruedasgreatpresumptioninustosupposethatnobodydoessobesidesourselves：andmuchmore，ifweshouldsaythattheGreatInventorofthismethod，andtheAuthorofsomanyotherwonderfuldiscoveries，neverkneworthoughtofwhattousappearssoplainandmanifest；
thathewhogaveussomuchlight，wasinthedarkhimself；thathewhoopenedourEyes，hadnosightofhisown。FormypartIcanneverconcurwithyouinthinkingthatIseefarther，orgobeyondSirIsaacNewton：Sedlongesequor，&；vestigiapronusadoro。Butifyouthinkfittopersistinassertingthatthisaffairofadoubleerrorisentirelyanewdiscoveryofyourown，whichSirIsaacandhisfollowersneverkneworthoughtof，Ihaveunquestionableevidencetoconvinceyouofthecontrary。Imustacquaintyouthereforewithwhatallhisfollowersarealreadyapprisedof，thattheseveryobjectionsofyourswerelongsinceforeseen，andclearlyandfullyremovedbySirIsaacNewton，inthefirstsectionofthefirstbookofhisPrincipia；
thegreaterpartofwhichsection，particularlythefirstandseventhLemma，andthatadmirableScholiumattheendofit，waswrittentothisveryendandpurposeonly，andtonootherintheworld。
Ihavenownomoretodo，butonlytoacquitmyselfofthepromiseImadeawhileago，torectifyamistakeyouarefallenintowithregardtoanotherofthegreatestmentheEnglishnationhasproduced。InordertowhichImustheretranscribethegreaterpartoftheCXXXVarticleofyourNewTheoryofVision。``AfterreiteratedendeavourstoapprehendthegeneralIdeaofaTriangle，Ihavefounditaltogetherincomprehensible。AndsurelyitanyonewereabletointroducethatIdeaintomyMind，itmustbetheAuthoroftheEssayconcerningHumanUnderstanding；He，whohassofardistinguishedhimselffromthegeneralityofWriters，bytheclearnessandsignificancyofwhathesays。LetusthereforeseehowthiscelebratedAuthordescribesthegeneral，orabstractIdeaofaTriangle。Itmustbe，sayshe，neitherOblique，norRectangular，neitherEquilateral，Equicrural，norScalenum；butallandnoneoftheseatonce。Ineffectitissomewhatimperfectthatcannotexist；anIdea，whereinsomepartsofseveraldifferentandinconsistentIdeasareputtogether。EssayonHumanUnderstanding。B。iv。C。7。S。9。ThisistheIdea，whichhethinksneedfulfortheEnlargementofKnowledge，whichisthesubjectofMathematicalDemonstration，andwithoutwhichwecouldnevercometoknowanygeneralPropositionconcerningTriangles。ThatAuthoracknowledgesitdothrequiresomepainsandskilltoformthisgeneralIdeaofaTriangle。Ibid。
Buthadhecalledtomindwhathesaysinanotherplace；towit，theIdeasofmixedModeswhereinanyinconsistentIdeasareputtogether，cannotsomuchasexistinthemind，i。e。beconceived。Vid。B。III。
C。10。S。33。Ibid。Isay，hadthisoccurredtohisThoughtsitisnotimprobablehewouldhaveowneditaboveallthePainsandSkillhewasmasterof，toformtheabove-mentionedIdeaofaTriangle，whichismadeupofmanifest，staringcontradictions。ThataManwhothoughtsomuch，andlaidsogreatastressonclearanddeterminateIdeas，shouldneverthelesstalkatthisrateseemsverysurprising。’’InthissectionyouplainlyaccuseMr。Lockeofcontradictinghimselfintwoseveralparticulars。Theabove-mentionedIdeaofaTriangle，sayyou，ismadeupofmanifest，staringcontradictions。YourepresentthetwofollowingpropositionsofMr。Lockeascontradictoryonetotheother。It，thegeneralIdeaofaTriangle，isanIdea，whereinsomepartsofseveraldifferentandinconsistentIdeasareputtogether。
Ideasofmixedmodes，whereinanyinconsistentideasareputtogether，cannotsomuchasexistintheMind。
Iproposetoclearupthesetwopoints，andtoshewthatinneitherofthemMr。Lockeisguiltyofcontradictinghimself：butfirst，inorderthereto，Imusttakeupalittleofyourtimeinconsideringthenotionofgeneral，orabstractIdeas。WhichpainsIamtheratherinclinedtotakebecause，thoughIhavecarefullyperusedwhatyouhavewrittenuponthissubject，Iamoneofthosewhostilladheretothevulgar，orratheruniversalerrorofallMankind，thatneitherGeometry，noranyothergeneralsciencecansubsistwithoutgeneralIdeas。
ThoughthewordsabstractorgeneralIdeasareindifferentlyusedbyWritersashavingthesamecommonsignification；yetasitmaybeameansofrenderingwhatIhavetosayuponthissubjectsomethingmoreintelligible，Ishallbegleavetomakeadistinctionbetweenthem，notasbeingdifferentinthemselves，butonlyinrespectofthemannerinwhichtheyarecommonlyformedorintroducedintothemind。
IshallconfinethenameofabstractIdeatothat，whichthemindformstoitselffromtheconsiderationofsomenumberofdifferentspecies，byabstractingfromthoseparticularIdeasinwhichthespeciesdifferfromoneanother，andretainingthoseinwhichtheyagree。
IshallcallthatageneralIdea，whichmaybeproducedinthemindwithoutanyconsideration，orevenknowledge，ofdifferentSpecies。
Anexamplewillmakethisveryplain。WhenMr。RayisforminghisMethodofPlants，heobservesthatMint，andSage，andLavender，andRosemary，andmanyotherPlants，besidestheirparticularcharacteristicksbywhichtheyaredistinguishedfromoneanother，havesomeothermarksinwhichtheyallagree；asintheirleavesgrowinginpairsoppositetoeachother，amonopetalouslabiateflower，withfourseedsgrowingatthebottomofit，andthoseinclosedinnoothervesselthantheperianthium。
ByjoiningtogetherthesecommonmarksheformshiscompoundIdeaofthatGenusofPlantswhichhecallsverticillate：whichfromhislayingaside，orabstractingfromallthepeculiardistinguishingmarksoftheseveralspecies，isproperlynamedanabstractIdea。
ButifMr。RaywillteachmeBotanybyhisMethod，hemusttakeadifferentcourse；hemustbeginwithmewherehehimselfended；hemustfirstintroduceintomymindthegeneralIdeaofaverticillateplant，andafterwarddescendtoparticularspecies。Hetellsmethataverticillateplantisonewhoseleavesgrowinpairsoppositetoeachother，andwhoseflowerismonopetalousandlabiate，withfourseedsatthebottomofit，andthoseinclosedonlyintheperianthium。ThisinmeisproperlycalledageneralIdea，becauseIshallfindittocomprehendalltheparticularspeciesofverticillateplants：butIhavenoreasontocallitanabstractIdea，becausenotknowingasyetanyoftheparticularspecies，ortheircharacteristickdifferences，Ihavenothingtoabstractfrom。
TheabstractIdeaisthatoftheMasterorPhilosopher；andthegeneralIdeathatoftheDisciple。Theformerrequires，asMr。Lockeobserves，somepainsandskilltoformit：thelatterdemandsneitherpainsnorskill，itneedsonlyalittleattentiontoconceiveit。
InlikemannerifapersonacquaintedwiththeseveralspeciesofTriangles，isfromtheconsiderationofthesetoformanIdeaofaTriangleingeneral；
hismethodwillbetoexaminetheseveralcompoundIdeasofthedifferentspeciesofTriangles，andtodistinguishbetweensuchpartsofthosecompoundIdeasasarethepeculiarcharacteristicksofeachspecies，andsuchpartsasarecommontoallofthemingeneral。ThenconnectingtheselasttogetherintoanewcompoundIdea，andabstractingfromalltherest，hewillhavetheabstractIdeaofaTriangle；whichisthatofaspacecomprehendedbythreerightlines，addifyouplease，containingthreeangles。
WhenhehasgotthisIdeahimself，itistheeasiestthingintheworld，togiveittoanother。LethimtakeaLearner，aBoy，suppose，whohasneverlearnedwhatatriangleis，muchlesswhatanyparticularspeciesofTriangleis，andtellhimaTriangleisaspacecomprehendedbythreerightlines：IsaythattheBoy，assoonasheunderstandsthemeaningofthesewords，willhaveacquiredthegeneralIdeaofaTriangle。Ifyoudoubtofit，shewhimarectangularTriangledrawnuponpaper，andaskhimwhatitis；hewillwithouthesitationtellyouitisaTriangle：afterwardsshewhimseparatelyalltheotherspeciesofTriangles，andyouwillfindheknowsthemeveryonetobeaTriangle。HisIdeaofaTrianglethereforeisgeneral，inasmuchasitsuitalltheparticularspecies。AndtheacquiringthisIdeaeitherabstract，orgeneral，inTeacherorScholar，seemstometobeattendedwithsolittledifficulty，thatIthinkMr。Lockehassaidfullenoughwhenhedeclaresthatthefirstrequiressomepainsandskilltoformit：anditistomesurprisingtohearaGentlemanofyourpenetrationprofessthatafterreiteratedendeavourstoapprehendthegeneralIdeaofaTriangle，youhavefounditaltogetherincomprehensible。
PutyourselfbutonceinthecaseofaLearner，endeavourtodivestyourmindofallyourpreconceivedGeometricalIdeas，andthenturntoEuclid’sdefinitions；andI’llventuretoassureyou，youwillfindnomoredifficultyinapprehendingthegeneralIdeaofaTriangle，thaninapprehendingtheIdeaofanobliqueangled，orofascaleneTriangle，oreventhatofanAnglealone；therebeingnoobjectionagainstthefirst，butwhatmaywithequalreasonbebroughtagainstanyoftheothers；aswilleasilyappeartohimthatconsiders，thatanangleingeneral，anobliqueangledTriangleingeneral，andascaleneTriangleingeneralcannowhereexistbutinIdeaonly，anymorethanaTriangleingeneral。
Havingpremisedthusmuchconcerningtheabstract，orgeneralIdeaofaTriangle，IcomenowtoexamineintoyourchargeagainstMr。Locke，andinthefirstplaceImusttakenoticethatthischargeisintroducedinanunfairandunjustmanner。IfanyonewereabletointroducethatIdeaintomymind；sayyou，itmustbetheAuthoroftheEssayconcerningHumanUnderstanding；&；c。LetusthereforeseehowthiscelebratedAuthordescribesthegeneral，orabstractIdeaofaTriangle。WouldnotanybodyimaginefromthesewordsthatMr。LockewereherepurposelydescribingthisIdea，inordertointroduceitintothemindofonewhohaditnotalready？Ifthatwerehisintention，itiscertainlyamostmiserabledescription；sincenopersonlivingwhodoesnotalreadyknowwhataTriangleis，caneverhavethatIdeaintroducedintohismindfromwhatMr。Lockeherelaysdown。AndyetthatIdeaisintroducedintothemindwithalltheeaseintheworldbywhathegivesustounderstandinanotherplace，[EssayonHum。Underst。B。II。C。31。
S。6。]thattheIdeaofaTriangleisthatofthreelines，includingaspace。Couldhepossiblytalksoclearlyinoneplace，andsocloudilyinanother，ifhisintentionwerethesameinboth？Isitnotplaintoanyonewhoattentivelyreadsthepassageyoureferto，thathisintentiontherewasnottodescribethegeneralIdeaofaTriangle，butonlytoshewfromtheseeminginconsistenciesinthatIdea，supposedtobealreadyknown，thatitrequiredsomepainsandskilltoformit，aswellasotherabstractIdeas？Observehiswords，``ForabstractIdeasarenotsoobviousoreasytochildren，ortheyetunexercisedmind，asparticularones。Iftheyseemsotogrownmen，’tisonlybecausebyconstantandfamiliarusetheyaremadeso。Forwhenwenicelyreflectuponthem，weshallfind，thatgeneralIdeasarefictionsandcontrivancesofthemind，thatcarrydifficultywiththem，anddonotsoeasilyofferthemselves，asweareapttoimagine。Forexample，DoesitnotrequiresomepainsandskilltoformthegeneralIdeaofaTriangle？（Whichyetisnoneofthemostabstract，comprehensiveanddifficult。）
Foritmustbeneitheroblique，norrectangle，neitherequilateral，equicrucialnorscalenon；butallandnoneoftheseatonce。Ineffect，itissomethingimperfect，thatcannotexist；anIdeawhereinsomepartsofseveraldifferentandinconsistentIdeasareputtogether。’’Wecomenowtothemanifest，staringcontradictions，containedinthisIdeaofaTriangle：thefirstofwhich，Isuppose，iscontainedinthesewords，allandnoneoftheseatonce。TheEnantiosis，Iconfess，isprettystrong：andyetthemeaningofitisplainlynomorethanthis，thatthegeneralIdeaofaTriangleisapartoftheIdeaofeveryspeciesofTriangleshereenumerated，butisnottheintireIdeaofanyoneofthem；iscommontothemall，andconfinedtonone。Itissomethingimperfectthatcannotexist，maypossiblybeanotherofyourcontradictions。
Itdoesnotappearsotome。ForeveryindividualTriangle，everyTrianglethatcanexist，mustbesomethingmorethanaspaceincludedbythreelines，itmustalsohavethecharacteristickmarkofsomeoneoftheparticularspeciesofTriangles；withoutwhichitwouldbeimperfect，itcouldnotexist，whichiswhatMr。LockeheresaysofaTriangleingeneral。
2。Butthegreatcontradictionofallseemstolieinthetwofollowingpropositions，whicharebroughttogetherfromdifferentpartsofMr。Locke’sworks，andsettostareoneanotherinthefacetodisgracetheirAuthor。
ItisanIdea，whereinsomepartsofseveraldifferentandinconsistentIdeasareputtogether。
Ideasofmixedmodes，whereinanyinconsistentIdeasareputtogether，cannotsomuchasexistinthemind。
Here，Sir，Istronglyapprehendyouarefallenintooneofthosetraps，whichthisGreatManwouldsometimesdiverthimselfwithsettingtocatchunwarycavillers，theHominesstolidos&；addepugnandumparatos，thatImentionedawhileago。Hadhisfirstpropositionrunthus，ItisanIdea，whereinseveraldifferentandinconsistentIdeasareputtogether，itwouldundoubtedlyhavebeencontradictorytothesecond。Butthatisnotthecase：prayobservethewordsofthiscautiousandaccurateWriter。
ItisanIdea，whereinSOMEPARTSOFseveraldifferentandinconsistentIdeasareputtogether。Now，weknowthattheseveralcompoundIdeasofarectangled，andoblique，andanacuteangledTrianglearedifferentandinconsistentonewithanother。Notwoofthemcanbeputtogethersoasjointlytoexistorbeconceivedinthemind。LikewisetheseveralcompoundIdeasofanequilateral，equicrural，andscalenetriangleareinconsistentwithoneanother。ButyetsomepartsofoneoftheseinconsistentIdeasarenotonlyconsistent，butareperfectlythesamewithsomepartsofanother。ToshewthisIbegleavetodividetwooftheseinconsistentIdeasintoseveralparts。ThecompoundIdeaThecompoundIdeaofarectangledTriangleofanacuteangledTrianglemaybedividedintomaybedividedintotheseparts。theseparts。
1。Aplainspace，1。Aplainspace，2。Comprehendedbyright2。Comprehendedbyrightlines，lines，3。Threeinnumber，3。Threeinnumber，4。Containingthreeangles，4。Containingthreeangles，5。Oneright，twoacute。5。Allacute。Thereis，wesee，nodifferencebetweenthefourfirstpartsofthecompoundIdeaofarectangledTriangle，andthefourfirstpartsofthatofanacuteangledTriangle：itisowingtothefifthpartaloneofeachIdea，thatthesetwoIdeasaredifferentandinconsistent。AndasitiseasytoseethatthesefourfirstpartsarethesameinallotherspeciesofTriangles；
andthatthesamefourpartsdocomposethegeneralIdeaofaTriangle；
itisplainthatthegeneralIdeaofaTriangleisanIdea，whereinSOME
PARTSofseveraldifferentandinconsistentIdeasareputtogether。
Thefirstthereforeofthetwopropositionsisquestionisundoubtedlytrue；andasthesepartsareinnowayinconsistentwithoneanother，itismanifestthatthesecondpropositionisnotcontradictory，oratallrepugnanttothefirst。
Icomenow，Sir，totakemyleaveofyou，andhopethatifanhonestzealfortruthinthefirstplace，andinthesecondforthereputationofthoseGentlementowhomIconceivethewholebodyofmankind，atleastImustacknowledgemyselftobehighlyindebted，hasgivenoccasionnotonlyofdifferingfromyou，butevenofreprehendingyouwiththeutmostfreedomwhereverIthoughtthetruthandyourbehaviourrequiredit；youwillnotimputethelibertyIhavetakentoanydisrespectforyourperson，whichIamanutterstrangerto，thoughIhaveaverygreatesteemandvalueforyouruncommonabilitiesandmanyofyourwritings，andamwithsincererespect，SIR，Yourmostobedient，HumbleServant，PHILALETHESCANTABRIGIENSIS。FINIS。