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；