ADefenceofFree—ThinkinginMathematicsInanswertoaPamphletofPhilalethesCantabrigiensis，intituled，GeometrynoFriendtoInfidelity，oraDefenceofSirISAACNEWTON，andtheBRITISHMathematicians。AlsoanAppendixconcerningMr。WALTON’SVindicationofthePrincipleofFluxionsagainsttheObjectionscontainedintheANALYST。
WHEREIN
ItisattemptedtoputthisControversyinsuchaLightasthateveryReadermaybeabletojudgethereof。ByGeorgeBerkeley1。WhenIreadyour`DefenceoftheBritishMathematicians，’
Icouldnot，Sir，butadmireyourcourageinassertingwithsuchundoubtingassurancethingssoeasilydisproved。Thistomeseemedunaccountable，tillIreflectedonwhatyousay（p。32），when，uponmyhavingappealedtoeverythinkingreader，whetheritbepossibletoframeanyclearconceptionofFluxions，youexpressyourselfinthefollowingmanner，``Pray，Sir，whoarethosethinkingreadersyouappealto？Aretheygeometricians，orpersonswhollyignorantofgeometry？Iftheformer，Ileaveittothem：
ifthelatter，Iask，Howwellaretheyqualifiedtojudgeofthemethodoffluxions？’’Itmustbeacknowledgedyouseembythisdilemmasecureinthefavourofonepartofyourreaders，andtheignoranceoftheother。
Iamneverthelesspersuadedtherearefairandcandidmenamongthemathematicians。
Andforthosewhoarenotmathematicians，Ishallendeavoursotounveilthismystery，andputthecontroversybetweenusinsuchalightasthateveryreaderofordinarysenseandreflectionmaybeacompetentjudgethereof。
2。Youexpressanextremesurpriseandconcern，``thatIshouldtakesomuchpainstodepreciateoneofthenoblestsciences，todisparageandtraduceasetoflearnedmen，whoselabourssogreatlyconducetothehonourofthisisland（p。5）；tolessenthereputationandauthorityofSirIsaacNewtonandhisfollowers，byshewingthattheyarenotsuchmastersofreasonastheyaregenerallypresumedtobe；andtodepreciatethesciencetheyprofess，bydemonstratingtotheworldthatitisnotofthatclearnessandcertaintyasiscommonlyimagined。’’Allwhich，youinsist，``appearsverystrangetoyouandtherestofthatfamousUniversity，whoplainlyseeofhowgreatusemathematicallearningistomankind。’’Henceyoutakeoccasiontodeclaimontheusefulnessofmathematicsintheseveralbranches，andthentoredoubleyoursurpriseandamazement（p。19and20）。ToallwhichdeclamationIreply，thatitisquitebesidethepurpose。For，Iallow，andalwayshaveallowed，itsfullclaimofmerittowhateverisusefulandtrueinthemathematics：butthatwhichisnotso，thelessitemploysmen’stimeandthoughtsthebetter。And，afterallyouhavesaidorcansay，Ibelievetheunprejudicedreaderwillthinkwithme，thatthingsobscurearenotthereforesacred；andthatitisnomoreacrimetocanvassanddetectunsoundprinciplesorfalsereasoningsinmathematicsthaninanyotherpartoflearning。
3。Youare，itseems，muchatalosstounderstandtheusefulness，ortendency，orprudenceofmyattempt。IthoughtIhadsufficientlyexplainedthisinthe`Analyst。’Butforyourfurthersatisfactionshallheretellyou，itisverywellknownthatseveralpersonswhoderideFaithandMysteriesinReligion，admitthedoctrineofFluxionsfortrueandcertain。Now，ifitbeshewnthatfluxionsarereallymostincomprehensiblemysteries，andthatthosewhobelievethemtobeclearandscientificdoentertainanimplicitfaithintheauthorofthatmethod：willnotthisfurnishafairargumentumadhominemagainstmenwhorejectthatverythinginreligionwhichtheyadmitinhumanlearning？Andisitnotaproperwaytoabatethepride，anddiscreditthepretensionsofthosewhoinsistuponclearideasinpointsoffaith，ifitbeshewnthattheydowithoutthemeveninscience。
4。Astomytimingthischarge；whynowandnotbefore，sinceIhadpublishedhintsthereofmanyyearsago？SurelyIamobligedtogivenoaccountofthis：ifwhathathbeensaidinthe`Analyst’
benotsufficient。SupposethatIhadnotleisure，orthatIdidnotthinkitexpedient，orthatIhadnomindtoit。Whenamanthinksfittopublishanything，eitherinmathematicsorinotherpartoflearning，whatavailsit，orindeedwhatrighthathanyonetoask，Whyatthisorthattime；
inthisorthatmanner；uponthisorthatmotive？LetthereaderjudgeifitsufficenotthatwhatIpublishistrue，andthatIhavearighttopublishsuchtruthswhenandhowIpleaseinafreecountry。
5。Idonotsaythatmathematicians，assuch，areinfidels；orthatgeometryisafriendtoinfidelity，whichyouuntrulyinsinuate，asyoudomanyotherthings；whenceyouraisetopicsforinvective。
ButIsaytherearecertainmathematicianswhoareknowntobeso；andthatthereareotherswhoarenotmathematicianswhoareinfluencedbyaregardfortheirauthority。Some，perhaps，wholiveintheUniversity，maynotbeapprisedofthis：buttheintelligentandobservingreader，wholivesintheworld，andisacquaintedwiththehumourofthetimesandthecharactersofmen，iswellawaretherearetoomanywhoderidemysteriesandyetadmirefluxions；whoyieldthatfaithtoameremortalwhichtheydenytoJesusChrist，whosereligiontheymakeittheirstudyandbusinesstodiscredit。Theowningthisisnottoownthatmenwhoreasonwellareenemiestoreligion，asyouwouldrepresentit：onthecontrary，Iendeavourtoshewthatsuchmenaredefectiveinpointofreasonandjudgement，andthattheydotheverythingtheywouldseemtodespise。
6。Thereare，Imakenodoubt，amongthemathematiciansmanysincerebelieversinJesusChrist：Iknowseveralsuchmyself：butIaddressedmy`Analyst’toaninfidel；and，onverygoodgrounds，Isupposedthat，besideshim，therewereotherderidersoffaithwhohadneverthelessaprofoundvenerationforfluxions：andIwaswillingtosetforththeinconsistenceofsuchmen。Iftherebenosuchthingasinfidelswhopretendtoknowledgeinthemodernanalysis，Iownmyselfmisinformed，andshallgladlybefoundinamistake；buteveninthatcase，myremarksonfluxionsarenotthelesstrue；norwillitfollowthatIhavenorighttoexaminethemonthefootofhumanscience，eventhoughreligionwerequiteunconcerned，andthoughIhadnoendtoservebuttruth。Butyouareveryangry（p。
13and14）thatIshouldenterthelistswithreasoninginfidels，andattackthemupontheirpretensionstoscience：andhenceyoutakeoccasionstoshewyourspleenagainsttheclergy。IwillnottakeuponmetosaythatIknowyoutobeaMinutePhilosopheryourself；butIknowtheMinutePhilosophersmakejustsuchcomplimentsasyoudotoourchurch，andarejustasangryasyoucanbeatanywhoundertaketodefendreligionbyreason。Ifweresolveallintofaith，theylaughatusandourfaith：andifweattempttoreason，theyareangryatus：theypretendwegooutofourprovince，andtheyrecommendtousablindimplicitfaith。Suchistheinconsistenceofouradversaries。Butitistobehopedtherewillneverbewantingmentodealwiththemattheirownweapons；andtoshewtheyarebynomeansthosemastersofreasonwhichtheywouldfainpassfor。
7。Idonotsay，asyouwouldrepresentme，thatwehavenobetterreasonforourreligionthanyouhaveforfluxions：butIsaythataninfidel，whobelievesthedoctrineoffluxions，actsaveryinconsistentpartinpretendingtorejecttheChristianreligionbecausehecannotbelievewhathedothnotcomprehend；orbecausehecannotassentwithoutevidence；orbecausehecannotsubmithisfaithtoauthority。Whethertherearesuchinfidels，Isubmittothejudgementofthereader。FormyownpartImakenodoubtofit，havingseensomeshrewdsignsthereofmyself，andhavingbeenverycrediblyinformedthereofbyothers。Nordoththischargeseemthelesscredible，foryourbeingsosensiblytouched，anddenyingitwithsomuchpassion。You，indeed，donotsticktoaffirm，thatthepersonswhoinformedmeare``apackofbase，profligate，andimpudentliars’’（p。27）。Howfarthereaderwillthinkfittoadoptyourpassions，Icannotsay；butIcantrulysay，thelatecelebratedMr。Addisonisoneofthepersonswhomyouarepleasedtocharacteriseinthesemodestandmannerlyterms。Heassuredmethattheinfidelityofacertainnotedmathematician，stillliving，wasoneprincipalreasonassignedbyawittymanofthosetimesforhisbeinganinfidel。NotthatIimaginegeometrydisposethmentoinfidelity：butthat，fromothercauses，suchaspresumption，ignorance，orvanity，likeothermengeometriciansalsobecomeinfidels，andthatthesupposedlightandevidenceoftheirsciencegainscredittotheirinfidelity。
8。Youreproachmewithcalumny，detraction，andartifice（p。15）。Yourecommendsuchmeansasareinnocentandjust，ratherthanthecriminalmethodoflesseningordetractingfrommyopponents（Ibid。）。
Youaccusemeoftheodiumtheologicum，theintemperatezealofdivines，thatIdostaresuperviasantiquas（p。13）；withmuchmoretothesameeffect。ForallwhichchargeIdependonthereader’scandour，thathewillnottakeyourword，butreadandjudgeforhimself。
Inwhichcasehewillbeabletodiscern（thoughheshouldbenomathematician）
howpassionateandunjustyourreproachesare，andhowpossibleitisforamantocryoutagainstcalumnyandpractiseitinthesamebreath。Consideringhowimpatientallmankindarewhentheirprejudicesarelookedinto，I
donotwondertoseeyourailandrageattherateyoudo。Butifyourownimaginationbestronglyshockedandmoved，youcannotthereforeconcludethatasincereendeavourtofreeascience，sousefulandornamentaltohumanlife，fromthosesubtleties，obscurities，andparadoxeswhichrenderitinaccessibletomostmen，willbethoughtacriminalundertakingbysuchasareintheirrightmind。MuchlesscanyouhopethatanillustriousSeminaryoflearnedmen，whichhathproducedsomanyfree—spiritedinquiriesaftertruth，willatonceenterintoyourpassions，anddegenerateintoanestofbigots。
9。Iobserveupontheinconsistencyofcertaininfidelanalysts。Iremarksomedefectsintheprinciplesofthemodernanalysis。
ItakethelibertydecentlytodissentfromSirIsaacNewton。Iproposesomehelpstoabridgethetroubleofmathematicalstudies，andrenderthemmoreuseful。Whatisthereinallthisthatshouldmakeyoudeclaimontheusefulnessofpracticalmathematics；thatshouldmoveyoutocryout，Spain，Inquisition，OdiumTheologicum？Bywhatfigureofspeechdoyouextendwhatissaidofthemodernanalysistomathematicsingeneral；
orwhatissaidofmathematicalinfidelstoallmathematicians；ortheconfutinganerrorinsciencetoburningorhangingtheauthors？Butitisnothingneworstrangethatmenshouldchoosetoindulgetheirpassions，ratherthanquittheiropinions，howabsurdsoever。Hencethefrightfulvisionsandtragicaluproarsofbigotedmen，bethesubjectoftheirbigotrywhatitwill。Averyremarkableinstanceofthisyougive（p。27），where，uponmyhavingsaidthatadeferencetocertainmathematicalinfidels，asIwascrediblyinformed，hadbeenonemotivetoinfidelity，youask，withnosmallemotion，``ForGod’ssakeareweinEnglandorinSpain？’’
``Isthisthelanguageofafamiliarwhoiswhisperinganinquisitor，&；c。？’’
Andthepagebeforeyouexclaiminthefollowingwords—``LetusburnorhangupallthemathematiciansinGreatBritain，orhalloothemobuponthemtotearthemtopieceseverymother’ssonofthem，TrosRutulusvefuat，laymenorclergymen，&；c。LetusdigupthebodiesofDr。
BarrowandSirIsaacNewton，andburnthemunderthegallows。’’
10。Thereaderneednotbeamathematiciantoseehowvainallthistragedyofyoursis。AndifhebeasthoroughlysatisfiedasIamthatthecauseoffluxionscannotbedefendedbyreason，hewillbeaslittlesurprisedasIamtoseeyoubetakeyourselftotheartsofallbigotedmen，raisingterrorandcallinginthepassionstoyourassistance。
Whetherthoserhetoricalflourishesabouttheinquisitionandthegallowsarenotquiteridiculous，Ileavetobedeterminedbythereader。Whowillalsojudge（thoughheshouldnotbeskilledingeometry）whetherIhavegiventheleastgroundsforthisandaworldofsuch—likedeclamation？
AndwhetherIhavenotconstantlytreatedthosecelebratedwriterswithallproperrespect，thoughItakethelibertyincertainpointstodifferfromthem？
11。AsIheartilyabhoraninquisitioninfaith，soIthinkyouhavenorighttoerectoneinscience。AtthetimeofwritingyourDefenceyouseemtohavebeenovercomewithpassion：but，nowyoumaybesupposedcool，Idesireyoutoreflectwhetheritbenotwroteinthetruespiritofaninquisitor？Whetherthisbecomesapersonsoexceedingdelicatehimselfuponthatpoint？Andwhetheryourbrethrentheanalystswillthinkthemselveshonouredorobligedbyyou，forhavingdefendedtheirdoctrineinthesamemannerasanydeclaimingbigotwoulddefendtransubstantiation？
Thesamefalsecolours，thesameintemperatesallies，andthesameindignationagainstcommonsense！
12。Inamatterofmerescience，whereauthorityhathnothingtodo，youconstantlyendeavourtooverbearmewithauthorities，andloadmewithenvy。IfIseeasophisminthewritingsofagreatauthor，and，incomplimenttohisunderstanding，suspecthecouldhardlybequitesatisfiedwithhisowndemonstration；thissetsyouondeclaimingforseveralpages。Itispompouslysetforth，asacriminalmethodofdetractingfromgreatmen，asaconcertedprojecttolessentheirreputation，asmakingthempassforimposters。IfIpublishmyfreethoughts，whichIhaveasmuchrighttopublishasanyotherman，itisimputedtorashness，andvanity，andtheloveofopposition。Thoughperhapsmylatepublication，ofwhathadbeenhintedtwenty—fiveyearsago，mayacquitmeofthischargeintheeyesofanimpartialreader。ButwhenIconsidertheperplexitiesthatbesetamanwhoundertakestodefendthedoctrineoffluxions，Icaneasilyforgiveyouranger。
13。Twosortsoflearnedmenthereare：onewhocandidlyseektruthbyrationalmeans。Theseareneveraversetohavetheirprincipleslookedinto，andexaminedbythetestofreason。Anothersortthereiswholearnbyroteasetofprinciplesandawayofthinkingwhichhappentobeinvogue。Thesebetraythemselvesbytheirangerandsurprise，whenevertheirprinciplesarefreelycanvassed。Butyoumustnotexpectthatyourreaderwillmakehimselfapartytoyourpassionsoryourprejudices。IfreelyownthatSirIsaacNewtonhathshewedhimselfanextraordinarymathematician，aprofoundnaturalist，apersonofthegreatestabilitiesanderudition。ThusfarIcanreadilygo；butIcannotgothelengthsthatyoudo。Ishallneversayofhimasyoudo，Vestigiapronusadoro（p。70）。ThissameadorationthatyoupaytohimIwillpayonlytotruth。
14。Youmay，indeed，yourselfbeanidolaterofwhomyouplease：butthenyouhavenorighttoinsultandexclaimatothermen，becausetheydonotadoreyouridol。GreatasSirIsaacNewtonwas，Ithinkhehath，onmoreoccasionsthanone，shewedhimselfnottobeinfallible。
Particularly，hisdemonstrationofthedoctrineoffluxionsItaketobedefective；andIcannothelpthinkingthathewasnotquitepleasedwithithimself。Andyetthisdothnothinderbutthatthemethodmaybeuseful，consideredasanartofinvention。You，whoareamathematician，mustacknowledgetherehavebeendiverssuchmethodsadmittedinmathematics，whicharenotdemonstrative。Such，forinstance，aretheinductionsofDr。Wallis，inhisArithmeticofInfinites，andsuchwhatHarriot，andafterhim，Descartes，havewroteconcerningtherootsofaffectedequations。Itwillnot，nevertheless，thencefollowthatthosemethodsareuseless；butonlythattheyarenottobeallowedofaspremisesinastrictdemonstration。
15。Nogreatnameuponearthshallevermakemeacceptthingsobscureforclear，orsophismsfordemonstrations。NormayyoueverhopetodetermefromfreelyspeakingwhatIfreelythink，bythoseargumentsadinvidiawhichateveryturnyouemployagainstme。Yourepresentyourself（p。52）asaman``whosehighestambitionisinthelowestdegreetoimitateSirIsaacNewton。’’Itmight，perhaps，havesuitedbetterwithyourappellationofPhilalethes，andbeenaltogetheraslaudable，ifyourhighestambitionhadbeentodiscovertruth。
Veryconsistentlywiththecharacteryougiveofyourself，youspeakofitasasortofcrime（p。70）tothinkitpossibleyoushouldever``seefarther，orgobeyondSirIsaacNewton。’’AndIampersuadedyouspeakthesentimentsofmanymorebesidesyourself。Butthereareotherswhoarenotafraidtosifttheprinciplesofhumanscience，whothinkitnohonourtoimitatethegreatestmaninhisdefects，whoeventhinkitnocrimetodesiretoknow，notonlybeyondSirIsaacNewton，butbeyondallmankind。Andwhoeverthinksotherwise，Iappealtothereaderwhetherhecanproperlybecalledaphilosopher。
16。BecauseIamnotguiltyofyourmeanidolatry，youinveighagainstmeasapersonconceitedofmyownabilities；notconsideringthatapersonoflessabilitiesmayknowmoreonacertainpointthanoneofgreater；notconsideringthatapurblindeye，inacloseandnarrowview，maydiscernmoreofathingthanamuchbettereyeinamoreextensiveprospect；notconsideringthatthisistofixaneplusultra，toputastoptoallfutureinquiries；lastly，notconsideringthatthisisinfact，somuchasinyoulies，convertingtherepublicoflettersintoanabsolutemonarchy，thatitisevenintroducingakindofphilosophicpoperyamongafreepeople。
17。Ihavesaid（andIventurestilltosay）thatafluxionisincomprehensible：thatsecond，third，andfourthfluxionsareyetmoreincomprehensible：thatitisnotpossibletoconceiveasimpleinfinitesimal：thatitisyetlesspossibletoconceiveaninfinitesimalofaninfinitesimal，andsoonward。[`Analyst，’sect。4，5，6，&；c。]
Whathaveyoutosayinanswertothis？Doyouattempttoclearupthenotionofafluxionoradifference？Nothinglikeit。Youonly``assureme（uponyourbareword）fromyourownexperience，andthatofseveralotherswhomyoucouldname，thatthedoctrineoffluxionsmaybeclearlyconceivedanddistinctlycomprehended；andthatifIampuzzledaboutitanddonotunderstandit，yetothersdo。’’Butcanyouthink，Sir，Ishalltakeyourword，whenIrefusetotakeyourmaster’s？
18。Uponthispointeveryreaderofcommonsensemayjudgeaswellasthemostprofoundmathematician。Thesimpleapprehensionofathingdefinedisnotmademoreperfectbyanysubsequentprogressinmathematics。Whatanymanevidentlyknows，heknowsaswellasyouorSirIsaacNewton。Andeveryonecanknowwhethertheobjectofthismethodbe（asyouwouldhaveusthink）clearlyconceivable。Tojudgeofthisnodepthofscienceisrequisite，butonlyabareattentiontowhatpassesinhisownmind。Andthesameistobeunderstoodofalldefinitionsinallscienceswhatsoever。Innoneofwhichcanitbesupposedthatamanofsenseandspiritwilltakeanydefinitionorprincipleontrust，withoutsiftingittothebottom，andtryinghowfarhecanorhecannotconceiveit。ThisisthecourseIhavetaken，andshalltake，howeveryouandyourbrethrenmaydeclaimagainstit，andplaceitinthemostinvidiouslight。
19。Itisusualwithyoutoadmonishmetolookoverasecondtime，toconsult，examine，weighthewordsofSirIsaac。
InanswertowhichIwillventuretosaythatIhavetakenasmuchpainsas（Isincerelybelieve）anymanlivingtounderstandthatgreatauthor，andtomakesenseofhisprinciples。Noindustry，norcaution，norattention，Iassureyou，havebeenwantingonmypart。Sothat，ifIdonotunderstandhim，itisnotmyfaultbutmymisfortune。Uponothersubjectsyouarepleasedtocomplimentmewithdepthofthoughtanduncommonabilities（p。
5and84）。ButIfreelyown，Ihavenopretencetothosethings。TheonlyadvantageIpretendtoisthatIhavealwaysthoughtandjudgedformyself。
And，asIneverhadamasterinmathematics，soIfairlyfollowedthedictatesofmyownmindinexaminingandcensuringtheauthorsIreaduponthatsubject，withthesamefreedomthatIuseduponanyother；takingnothingontrust，andbelievingthatnowriterwasinfallible。Andamanofmoderateparts，whotakesthispainfulcourseinstudyingtheprinciplesofanyscience，maybesupposedtowalkmoresurelythanthoseofgreaterabilities，whosetoutwithmorespeedandlesscare。
20。WhatIinsistonis，thattheideaofafluxion，simplyconsidered，isnotatallimprovedoramendedbyanyprogress，thougheversogreat，intheanalysis：neitherarethedemonstrationsofthegeneralrulesofthatmethodatallclearedupbyapplyingthem。Thereasonofwhichis，because，inoperatingorcalculating，mendonotreturntocontemplatetheoriginalprinciplesofthemethod，whichtheyconstantlypresuppose，butareemployedinworking，bynotesandsymbolsdenotingthefluxionssupposedtohavebeenatfirstexplained，andaccordingtorulessupposedtohavebeenatfirstdemonstrated。ThisIsaytoencouragethosewhoarenottoofargoneinthesestudies，touseintrepidlytheirownjudgement，withoutablindorameandeferencetothebestofmathematicians，whoarenomorequalifiedthantheyaretojudgeofthesimpleapprehension，ortheevidenceofwhatisdeliveredinthefirstelementsofthemethod；
menbyfurtherandfrequentuseorexercisebecomingonlymoreaccustomedtothesymbolsandrules，whichdothnotmakeeithertheforegoingnotionsmoreclear，ortheforegoingproofsmoreperfect。Everyreaderofcommonsense，thatwillbutusehisfaculties，knowsaswellasthemostprofoundanalystwhatideaheframesorcanframeofvelocitywithoutmotion，orofmotionwithoutextension，ofmagnitudewhichisneitherfiniteorinfinite，orofaquantityhavingnomagnitudewhichisyetdivisible，ofafigurewherethereisnospace，ofproportionbetweennothings，orofarealproductfromnothingmultipliedbysomething。Heneednotbefargoneingeometrytoknowthatobscureprinciplesarenottobeadmittedindemonstration；
thatifamandestroyshisownhypothesis，heatthesametimedestroyswhatwasbuiltuponit：thaterrorinthepremises，notrectified，mustproduceerrorintheconclusion。
21。Inmyopinionthegreatestmenhavetheirprejudices。
Menlearntheelementsofsciencefromothers：andeverylearnerhathadeferencemoreorlesstoauthority，especiallytheyounglearners，fewofthatkindcaringtodwelllonguponprinciples，butincliningrathertotakethemupontrust：andthingsearlyadmittedbyrepetitionbecomefamiliar：andthisfamiliarityatlengthpassethforevidence。Nowtomeitseemstherearecertainpointstacitlyadmittedbymathematicianswhichareneitherevidentnortrue。Andsuchpointsorprinciplesevermixingwiththeirreasoningsdoleadthemintoparadoxesandperplexities。Ifthegreatauthorofthefluxionarymethodwereearlyimbuedwithsuchnotionsitwouldonlyshewhewasaman。Andif，byvirtueofsomelatenterrorinhisprinciples，amanbedrawnintofallaciousreasonings，itisnothingstrangethatheshouldtakethemfortrue：andnevertheless，if，whenurgedbyperplexitiesanduncouthconsequences，anddriventoartsandshifts，heshouldentertainsomedoubtthereof，itisnomorethanonemaynaturallysupposemightbefallagreatgeniusgrapplingwithaninsuperabledifficulty：
whichisthelightinwhichIhaveplacedSirIsaacNewton。[`Analyst，’
sect。18。]HereuponyouarepleasedtoremarkthatIrepresentthegreatauthornotonlyasaweakbutasanillman，asadeceiverandanimpostor。
Thereaderwilljudgehowjustly。
22。Astotherestofyourcolouringsandglosses，yourreproachesandinsultsandoutcries，Ishallpassthemover，onlydesiringthereadernottotakeyourword，butreadwhatIhavewritten，andhewillwantnootheranswer。Ithathbeenoftenobservedthattheworstcauseproduceththegreatestclamour；andindeedyouaresoclamorousthroughoutyourdefencethatthereader，althoughheshouldbenomathematician，providedheunderstandscommonsense，andhathobservedthewaysofmen，willbeapttosuspectthatyouareinthewrong。Itshouldseem，therefore，thatyourbrethrentheanalystsarebutlittleobligedtoyouforthisnewmethodofdeclaiminginmathematics。WhethertheyaremoreobligedbyyourreasoningIshallnowexamine。
23。Youaskme（p。32）whereIfindSirIsaacNewtonusingsuchexpressionsasthevelocitiesofvelocities，thesecond，third，andfourthvelocities，&；c。Thisyousetforthasapiousfraudandunfairrepresentation。Ianswer，thatifaccordingtoSirIsaacNewtonafluxionbethevelocityofanincrement，thenaccordingtohimImaycallthefluxionofafluxionthevelocityofavelocity。Butforthetruthoftheantecedentseehis`IntroductiontotheQuadratureofCurves，’wherehisownwordsare，Motuumvelincrementorumvelocitatesnominandofluxiones。
SeealsothesecondlemmaofthesecondbookofhisMathematicalPrinciplesofNaturalPhilosophy，whereheexpressethhimselfinthefollowingmanner：Velocitatesincrementorumacdecrementorumquasetiam，motus，mutationes，etfluxionesquantitatumnominarelicet。Andthatheadmitsfluxionsoffluxions，orsecond，third，fourthfluxions，&；c。，seehisTreatiseoftheQuadratureofCurves。Iasknow，Isitnotplainthatifafluxionbeavelocity，thenthefluxionofafluxionmay，agreeablythereunto，becalledthevelocityofavelocity？Inlikemanner，ifbyafluxionismeantanascentaugment，willitnotthenfollowthatthefluxionofafluxionorsecondfluxionisthenascentaugmentofanascentaugment？
Cananythingbeplainer？Letthereadernowjudgewhoisunfair。
24。Ihadobservedthatthegreatauthorhadproceededillegitimately，inobtainingthefluxionormomentoftherectangleoftwoflowingquantities；andthathedidnotfairlygetridoftherectangleofthemoments。Inanswertothis，youallegethattheerrorarisingfromtheomissionofsuchrectangle（allowingittobeanerror）issosmallthatitisinsignificant。Thisyoudwelluponandexemplifytonootherpurposebuttoamuseyourreaderandmisleadhimfromthequestion；whichintruthisnotconcerningtheaccuracyofcomputingormeasuringinpractice，butconcerningtheaccuracyofthereasoninginscience。Thatthiswasreallythecase，andthatthesmallnessofthepracticalerrornowiseconcernsit，mustbesoplaintoanyonewhoreadsthe`Analyst’thatIwonderhowyoucouldbeignorantofit。
25。YouwouldfainpersuadeyourreaderthatI
makeanabsurdquarrelagainsterrorsofnosignificancyinpractice，andrepresentmathematiciansasproceedingblindfoldintheirapproximations，inallwhichIcannothelpthinkingthereisonyourparteithergreatignoranceorgreatdisingenuity。Ifyoumeantodefendthereasonablenessanduseofapproximationsorofthemethodofindivisibles，Ihavenothingtosay。Butthenyoumustrememberthisisnotthedoctrineoffluxions：
itisnoneofthatanalysiswithwhichIamconcerned。ThatIamfarfromquarrellingatapproximationsingeometryismanifestfromthethirty—thirdandfifty—thirdqueriesinthe`Analyst。’Andthatthemethodoffluxionspretendstosomewhatmorethanthemethodofindivisiblesisplain；becauseSirIsaacdisclaimsthismethodasnotgeometrical。[SeetheScholiumattheendofthefirstsection。Lib。i。，`Phil。Nat。Princip。Math。’]Andthatthemethodoffluxionsissupposedaccurateingeometricalrigourismanifesttowhoeverconsiderswhatthegreatauthorwritesaboutit；
especiallyinhis`IntroductiontotheQuadratureofCurves，’wherehesaith，Inrebusmathematiciserroresquamminiminonsuntcontemnendi。
Whichexpressionyouhaveseenquotedinthe`Analyst，’andyetyouseemignorantthereof，andindeedoftheveryendanddesignofthegreatauthorofthishisinventionoffluxions。
26。Asoftasyoutalkoffinitequantitiesinconsiderableinpractice，SirIsaacNewtondisownsyourapology。Cave，saithhe，intellexerisfinitas。And，althoughquantitieslessthansensiblemaybeofnoaccountinpractice，yetnoneofyourmasters，notwillevenyouyourself，venturetosaythattheyareofnoaccountintheoryandinreasoning。Theapplicationingrosspracticeisnotthepointquestioned，buttherigourandjustnessofthereasoning。Anditisevidentthat，bethesubjecteversolittle，oreversoinconsiderable，thisdothnothinderbutthatapersontreatingthereofmaycommitverygreaterrorsinlogic；
whichlogicalerrorsareinnowisetobemeasuredbythesensibleorpracticalinconveniencesthencearising，which，perchance，maybenoneatall。Itmustbeownedthat，afteryouhavemisledandamusedyourlessqualifiedreader（asyoucallhim），youreturntotherealpointincontroversy，andsetyourselftojustifySirIsaac’smethodofgettingridoftheabove—mentionedrectangle。AndhereImustintreatthereadertoobservehowfairlyyouproceed。
27。Firstthenyouaffirm（p。44），``thatneitherinthedemonstrationoftheruleforfindingthefluxionoftherectangleoftwoflowingquantities，norinanythingprecedingorfollowingit，isanymention，somuchasonce，madeoftheincrementoftherectangleofsuchflowingquantities。’’NowIaffirmthedirectcontrary。For，intheverypassagebyyouquotedinthissamepage，fromthefirstcaseofthesecondlemmaofthesecondbookofSirIsaac’sPrinciples，beginningwithRectangulumquodvismotuperpetuoauctum，andendingwithigiturlaterumincrementistotisaandbgeneraturrectanguliincrementumaB
bA。Q。E。D。inthisverypassage，Isay，isexpressmentionmadeoftheincrementofsuchrectangle。Asthisismatteroffact，Ireferittothereader’sowneyes。Ofwhatrectanglehaveweheretheincrement？
Isitnotplainlyofthatwhosesideshaveaandbfortheirincrementatota，thatis，ofAB。Letanyreaderjudgewhetheritbenotplainfromthewords，thesense，andthecontext，thatthegreatauthorintheendofhisdemonstrationunderstandshisincrementumasbelongingtotherectangulumquodvisatthebeginning。Isnotthesamealsoevidentfromtheverylemmaitselfprefixedtothedemonstration？
Thesensewhereofis（astheauthorthereexplainsit），thatifthemomentsoftheflowingquantitiesAandBarecalledaandb，thenthemomentumvelmutatiogenitirectanguliABwillbeaBbA。Eitherthereforetheconclusionofthedemonstrationisnotthethingwhichwastobedemonstrated，ortherectanguliincrementumaBbAbelongstotherectangleAB。
28。Allthisissoplainthatnothingcanbemoreso；andyetyouwouldfainperplexthisplaincasebydistinguishingbetweenanincrementandamoment。Butitisevidenttoeveryonewhohasanynotionofdemonstrationthattheincrementumintheconclusionmustbethemomentuminthelemma；andtosupposeitotherwiseisnocredittotheauthor。Itisineffectsupposinghimtobeonewhodidnotknowwhathewoulddemonstrate。ButletushearSirIsaac’sownwords：Earum（quantitatumscilicetfluentium）incrementaveldecrementamomentaneasubnominemomentorumintelligo。Andyouobserveyourselfthatheuseththewordmomenttosignifyeitheranincrementordecrement。Hence，withanintentiontopuzzleme，youproposetheincrementanddecrementofAB，andaswhichoftheseIwouldcallthemoment？Thecaseyousayisdifficult。Myanswerisveryplainandeasy，towit，Eitherofthem。
You，indeed，makeadifferentanswer；andfromtheauthor’ssayingthatbyamomentheunderstandseitherthemomentaneousincrementordecrementoftheflowingquantities，youwouldhaveusconclude，byaverywonderfulinference，thathismomentisneithertheincrementnordecrementthereof。
Woulditnotbeasgoodaninference，becauseanumberiseitheroddoreven，toconcludeitisneither？Cananyonemakesenseofthis？Orcanevenyourselfhopethatthiswillgodownwiththereader，howlittlesoeverqualified？Itmustbeowned，youendeavourtointrudethisinferenceonhim，ratherbymirthandhumourthanbyreasoning。Youraremerry，Isay，and（p。46）representthetwomathematicalquantitiesaspleadingtheirrights，astossingupcrossandpile，asdisputingamicably。Youtalkoftheirclaimingpreference，theiragreeing，theirboyishness，andtheirgravity。Andafterthisingeniousdigressionyouaddressmeinthefollowingwords—Believeme，thereisnoremedy，youmustacquiesce。ButmyansweristhatIwillneitherbelieveyounoracquiesce；thereisaplainremedyincommonsense；and，topreventsurprise，Idesirethereaderalwaystokeepthecontrovertedpointinview，toexamineyourreasons，andbecautioushowhetakesyourword，butmostofallwhenyouarepositive，oreloquent，ormerry。
29。Apageortwoafter，youverycandidlyrepresentyourcasetobethatofanassbetweentwobottlesofhay：itisyourownexpression。ThecauseofyourperplexityisthatyouknownotwhetherthevelocityofABincreasing，orofABdecreasingistobeesteemedthefluxion，orproportionaltothemomentoftherectangle。Myownopinion，agreeablytowhathathbeenpremised，isthateithermaybedeemedthefluxion。Butyoutellus（p。49）``thatyouthink，thevenerableghostofSirIsaacNewtonwhispersyou，thevelocityyouseekforisneithertheonenortheotherofthese，butitisthevelocitywhichtheflowingrectanglehathnotwhileitisgreaterorlessthanAB，butatthatveryinstantoftimethatitisAB。’’Formypart，intherectangleABconsideredsimplyinitself，withouteitherincreasingordiminishing，Icanconceivenovelocityatall。Andifthereaderisofmyownmind，hewillnottakeeitheryourword，oreventhewordofaghost，howvenerablesoever，forvelocitywithoutmotion。Youproceedandtellusthat，inlikemanner，themomentoftherectangleisneitheritsincrementordecrement。
Thisyouwouldhaveusbelieveontheauthorityofhisghost，indirectoppositiontowhatSirIsaachimselfassertedwhenalive。Incrementa（saithhe）veldecrementamomentaneasubnominemomentorumintelligo：
itautincrementapromomentisaddititiisseuaffirmativis，acdecrementaprosubductitiisseunegativishabeantur。[`Princip。Phil。Nat。，’lib。
ii，lem。ii。]Iwillnotinyourstylebidthereaderbelieveme，butbelievehiseyes。
30。Tomeitverilyseemsthatyouhaveundertakenthedefenceofwhatyoudonotunderstand。Tomendthematter，yousay，``youdonotconsiderABaslyingateitherextremityofthemoment，butasextendedtothemiddleofit；ashavingacquiredtheonehalfofthemoment，andasbeingabouttoacquiretheother；orashavinglostonehalfofit，andbeingabouttolosetheother。’’Now，inthenameoftruth，Ientreatyoutotellwhatthismomentis，tothemiddlewhereoftherectangleisextended？Thismoment，Isay，whichisacquired，whichislost，whichiscutintwo，ordistinguishedintohalves？Isitafinitequantity，oraninfinitesimal，oramerelimit，ornothingatall？Takeitinwhatsenseyouwill，Icannotmakeyourdefenceeitherconsistentorintelligible。For，ifyoutakeitineitherofthetwoformersenses，youcontradictSirIsaacNewton。And，ifyoutakeitineitherofthelatter，youcontradictcommonsense；itbeingplain，thatwhathathnomagnitude，orisnoquantity，cannotbedivided。AndhereImustentreatthereadertopreservehisfullfreedomofmindentire，andnotweaklysufferhisjudgementtobeoverbornebyyourimaginationandyourprejudices，bygreatnamesandauthorities，byghostsandvisions，andaboveallbythatextremesatisfactionandcomplacencywithwhichyouutteryourstrangeconceits；
ifwordswithoutameaningmaybecalledso。Afteryouhavegiventhisunintelligibleaccount，youaskwithyouraccustomedair，``Whatsayyou，Sir？IsthisajustandlegitimatereasonforSirIsaac’sproceedingashedid？Ithinkyoumustacknowledgeittobeso。’’But，alas！Iacknowledgenosuchthing。Ifindnosenseorreasoninwhatyousay。Letthereaderfinditifhecan。
31。Inthenextplace（p。50），youchargemewithwantofcaution。``Inasmuch（sayyou）asthatquantitywhichSirIsaacNewton，throughhiswholelemma，andalltheseveralcasesofit，constantlycallsamoment，withoutconfiningittobeeitheranincrementordecrement，isbyyouinconsideratelyandarbitrarily，andwithoutanyshadowofreasongiven，supposedanddeterminedtobeanincrement。’’TowhichchargeIreply，thatitisasuntrueasitisperemptory。Forthat，intheforegoingcitationfromthefirstcaseofSirIsaac’slemma，heexpresslydeterminesittobeanincrement。And，asthisparticularinstanceorpassagewasthatwhichIobjectedto，itwasreasonableandproperformetoconsiderthemomentinthesamelight。But，takeitincrementordecrementasyouwill，theobjectionsstilllie，andthedifficultiesareequallyinsuperable。
Youthenproceedtoextolthegreatauthorofthefluxionarymethod，andtobestowsomebrusqueriesuponthosewhounadvisedlydaretodifferfromhim。ToallwhichIshallgivenoanswer。
32。Afterwardstoremove（asyousay）allscrupleanddifficultyaboutthisaffair，youobservethatthemomentoftherectangledeterminedbySirIsaacNewton，andtheincrementoftherectangledeterminedbymeareperfectlyandexactlyequal，supposingaandbtobediminishedadinfinitum：and，forproofofthis，yourefertothefirstlemmaofthefirstsectionofthefirstbookofSirIsaac’sprinciples。Ianswerthatifaandbarerealquantitiesthenabissomething，andconsequentlymakesarealdifference：
butiftheyarenothing，thentherectangleswhereoftheyarecoefficientsbecomenothinglikewise：andconsequentlythemomentumorincrementum，whetherSirIsaac’sormine，areinthatcasenothingatall。Asfortheabove—mentionedlemma，whichyoureferto，andwhichyouwishIhadconsultedsooner，bothformyownsakeandforyours；ItellyouIhadlongsinceconsultedandconsideredit。ButIverymuchdoubtwhetheryouhavesufficientlyconsideredthatlemma，itsdemonstration，anditsconsequences。For，howeverthatwayofreasoningmaydointhemethodofexhaustions，wherequantitieslessthanassignableareregardedasnothing：yet，forafluxionistwritingaboutmomentumstoarguethatquantitiesmustbeequalbecausetheyhavenoassignabledifferenceseemsthemostinjudiciousstepthatcouldbetaken：itisdirectlydemolishingtheverydoctrineyouwoulddefend。For，itwillthencefollowthatallhomogeneousmomentumsareequal，andconsequentlythevelocities，mutations，orfluxions，proportionalthereto，arealllikewiseequal。Thereis，therefore，onlyoneproportionofequalitythroughout，whichatonceoverthrowsthewholesystemyouundertaketodefend。Yourmoments（Isay）notbeingthemselvesassignablequantities，theirdifferencescannotbeassignable：and，ifthisbetrue，bythatwayofreasoningitwillfollow，theyareallequal；uponwhichsuppositionyoucannotmakeonestepinthemethodoffluxions。Itappearsfromhence，howunjustlyyoublameme（p。32）foromittingtogiveanyaccountofthatfirstsectionofthefirstbookofthe`Principia，’wherein（yousay）thefoundationofthemethodoffluxionsisgeometricallydemonstratedandlargelyexplained，anddifficultiesandobjectionsagainstitareclearlysolved。Allwhichissofarfrombeingtruethattheveryfirstandfundamentallemmaofthatsectionisincompatiblewithandsubversiveofthedoctrineoffluxions。And，indeed，whoseesnotthatademonstrationadabsurdummoreveterum，proceedingonasuppositionthateverydifferencemustbesomegivenquantity，cannotbeadmittedin，orconsistwith，amethodwhereinquantities，lessthananygiven，aresupposedreallytoexist，andbecapableofdivision？
33。Thenextpointyouundertaketodefendisthatmethodforobtainingaruletofindthefluxionofanypowerofaflowingquantity，whichisdeliveredinhis`IntroductiontotheQuadratures，’