andconsideredinthe`Analyst。’Andherethequestionbetweenusis，whetherIhaverightlyrepresentedthesenseofthosewordsevanescantjamaugmentailla，inrenderingthem，``lettheincrementsvanish，’’i。e。lettheincrementsbenothing，orlettherebenoincrements？Thisyoudeny；but，asyourmanneris，insteadofgivingareasonyoudeclaim。I，onthecontrary，affirm，theincrementsmustbeunderstoodtobequitegone，andabsolutelynothingatall。Myreasonis，becausewithoutthatsuppositionyoucanneverbringthequantityorexpressiondownto，theverythingaimedatbysupposingtheevanescence。Saywhetherthisbenotthetruthofthecase？Whethertheformerexpressionisnottobereducedtothelatter？
Andwhetherthiscanpossiblybedonesolongasoisarealquantity？
Icannotindeedsayyouarescrupulousaboutyouraffirmations，andyetIbelievethatevenyouwillnotaffirmthis；itbeingmostevident，thattheproductoftworealquantitiesissomethingreal；andthatnothingrealcanberejectedeitheraccordingtotheofgeometry，oraccordingtoSirIsaac’sownPrinciples；forthetruthofwhichIappealtoallwhoknowanythingofthesematters。Further，byevanescentmusteitherbemeant，letthem（theincrements）vanishandbecomenothing，intheobvioussense，orletthembecomeinfinitelysmall。ButthatthislatterisnotSirIsaac’ssenseisevidentfromhisownwordsintheverysamepage，thatis，inthelastofhis`IntroductiontotheQuadratures，’whereheexpresslysaith，voluiostenderequodinmethodofluxionumnonopussitfigurasinfiniteparvasingeometriamintroducere。Uponthewhole，youseemtohaveconsideredthisaffairsoverysuperficiallyasgreatlytoconfirmmeintheopinionyouaresoangrywith，towit，thatSirIsaac’sfollowersaremuchmoreeagerinapplyinghismethodthanaccurateinexamininghisprinciples。Youraiseadustaboutevanescentaugments，whichmayperhapsamuseandamazeyourreader，butIammuchmistakenifiteverinstructsorenlightenshim。For，tocometothepoint，thoseevanescentaugmentseitherarerealquantities，ortheyarenot。Ifyousaytheyare；Idesiretoknowhowyougetridoftherejectaneousquantity？Ifyousaytheyarenot；youindeedgetridofthosequantitiesinthecompositionwhereoftheyarecoefficients；butthenyouareofthesameopinionwithme，whichopinionyouarepleasedtocall（p。58）``amostpalpable，inexcusable，andunpardonableblunder，’’
althoughitbeatruthmostpalpablyevident。
34。Nothing，Isay，canbeplainertoanyimpartialreaderthanthat，bytheevanescenceofaugmentsintheabove—citedpassage，SirIsaacmeanstheirbeingactuallyreducedtonothing。But，toputitoutofalldoubtthatthisisthetruth，andtoconvinceevenyou，whoshewsolittledispositiontobeconvinced，Idesireyoutolookintohis``AnalysisperAequationesInfinitas’’（p。20），where，inhispreparationfordemonstratingthefirstruleforthesquaringofsimplecurves，youwillfindthat，onaparalleloccasion，speakingofanaugmentwhichissupposedtovanish，heinterpretsthewordevanescerebyessenihil。Nothingcanbeplainerthanthis，whichatoncedestroysyourdefence。Andyet，plainasitis，Idespairofmakingyouacknowledgeit；