27。As，ontheonehand，itwereabsurdtogetridofobysaying，Letmecontradictmyself；letmesubvertmyownhypothesis；letmetakeitforgrantedthatthereisnoincrement，atthesametimethatIretainaquantitywhichIcouldneverhavegotatbutbyassuminganincrement：so，ontheotherhand，itwouldbeequallywrongtoimaginethatinageometricaldemonstrationwemaybeallowedtoadmitanyerror，thougheversosmall，orthatitispossible，inthenatureofthings，anaccurateconclusionshouldbederivedfrominaccurateprinciples。Thereforeocannotbethrownoutasaninfinitesimal，orupontheprinciplethatinfinitesimalsmaybesafelyneglected；butonlybecauseitisdestroyedbyanequalquantitywithanegativesign，whenceo-poisequaltonothing。Andasitisillegitimatetoreduceanequation，bysubductingfromonesideaquantitywhenitisnottobedestroyed，orwhenanequalquantityisnotsubductedfromtheothersideoftheequation：soitmustbeallowedaverylogicalandjustmethodofarguingtoconcludethatiffromequalseithernothingorequalquantitiesaresubductedtheyshallstillremainequal。Andthisisatruereasonwhynoerrorisatlastproducedbytherejectingofo。Whichthereforemustnotbeascribedtothedoctrineofdifferences，orinfinitesimals，orevanescentquantities，ormomentums，orfluxions。
28。Supposethecasetobegeneral，andthatisequaltotheareaABCwhencebythemethodoffluxionstheordinateisfound，whichweadmitfortrue，andshallinquirehowitisarrivedat。Nowifwearecontenttocomeattheconclusioninasummaryway，bysupposingthattheratioofthefluxionsofxandisfound[Sect。13。]tobe1and，andthattheordinateoftheareaisconsideredasitsfluxion，weshallnotsoclearlyseeourway，orperceivehowthetruthcomesout，thatmethodaswehaveshewedbeforebeingobscureandillogical。Butifwefairlydelineatetheareaanditsincrement，anddividethelatterintotwopartsBCFDandCFH，[Seethefigureinsect。26。]andproceedregularlybyequationsbetweenthealgebraicalandgeometricalquantities，thereasonofthethingwillplainlyappear。ForasisequaltotheareaABC，soistheincrementofequaltotheincrementofthearea，i。e。toBDHC；thatistosayAndonlythefirstmembersoneachsideoftheequationbeingretained，=BDFC：anddividingbothsidesbyoorBD，weshallget=BC。AdmittingthereforethatthecurvilinearspaceCFHisequaltotherejectaneousquantityandthatwhenthisisrejectedononeside，thatisrejectedontheother，thereasoningbecomesjustandtheconclusiontrue。AnditisallonewhatevermagnitudeyouallowtoBD，whetherthatofaninfinitesimaldifferenceorafiniteincrementeversogreat。Itisthereforeplainthatthesupposingtherejectaneousalgebraicalquantitytobeaninfinitelysmallorevanescentquantity，andthereforetobeneglected，musthaveproducedanerror，haditnotbeenforthecurvilinearspacesbeingequalthereto，andatthesametimesubductedfromtheotherpartorsideoftheequation，agreeablytotheaxiom，Iffromequalsyousubductequals，theremainderswillbeequal。Forthosequantitieswhichbytheanalystsaresaidtobeneglected，ormadetovanish，areinrealitysubducted。Ifthereforetheconclusionbetrue，itisabsolutelynecessarythatthefinitespaceCFHbeequaltotheremainderoftheincrementexpressedbyequal，Isay，tothefiniteremainderofafiniteincrement。
29。Therefore，bethepowerwhatyouplease，therewillariseononesideanalgebraicalexpression，ontheotherageometricalquantity，eachofwhichnaturallydividesitselfintothreemembers。Thealgebraicalorfluxionaryexpression，intoonewhichincludesneithertheexpressionoftheincrementoftheabscissanorofanypowerthereof；anotherwhichincludestheexpressionoftheincrementitself；andthethirdincludingtheexpressionofthepowersoftheincrement。Thegeometricalquantityalsoorwholeincreasedareaconsistsofthreepartsormembers，thefirstofwhichisthegivenarea；thesecondarectangleundertheordinateandtheincrementoftheabscissa；thethirdacurvilinearspace。And，comparingthehomologousorcorrespondentmembersonbothsides，wefindthatasthefirstmemberoftheexpressionistheexpressionofthegivenarea，sothesecondmemberoftheexpressionwillexpresstherectangleorsecondmemberofthegeometricalquantity，andthethird，containingthepowersoftheincrement，willexpressthecurvilinearspace，orthirdmemberofthegeometricalquantity。Thishintmayperhapsbefurtherextended，andappliedtogoodpurpose，bythosewhohaveleisureandcuriosityforsuchmatters。TheuseImakeofitistoshew，thattheanalysiscannotobtaininaugmentsordifferences，butitmustalsoobtaininfinitequantities，betheyeversogreat，aswasbeforeobserved。
30。Itseemstherefore，uponthewhole，thatwemaysafelypronouncetheconclusioncannotberight，ifinordertheretoanyquantitybemadetovanish，orbeneglected，exceptthateitheroneerrorisredressedbyanother；orthat，secondly，onthesamesideofanequationequalquantitiesaredestroyedbycontrarysigns，sothatthequantitywemeantorejectisfirstannihilated；or，lastly，thatfromoppositesidesequalquantitiesaresubducted。Andthereforetogetridofquantitiesbythereceivedprinciplesoffluxionsorofdifferencesisneithergoodgeometrynorgoodlogic。Whentheaugmentsvanish，thevelocitiesalsovanish。Thevelocitiesorfluxionsaresaidtobeprimoandultimo，astheaugmentsnascentandevanescent。Takethereforetheratiooftheevanescentquantities，itisthesamewiththatofthefluxions。Itwillthereforeanswerallintentsaswell。Whythenarefluxionsintroduced？Isitnottoshunorrathertopalliatetheuseofquantitiesinfinitelysmall？Butwehavenonotionwherebytoconceiveandmeasurevariousdegreesofvelocitybesidesspaceandtime；or，whenthetimesaregiven，besidesspacealone。Wehaveevennonotionofvelocityprescindedfromtimeandspace。Whenthereforeapointissupposedtomoveingiventimes，wehavenonotionofgreaterorlesservelocities，orofproportionsbetweenvelocities，butonlyoflongerandshorterlines，andofproportionsbetweensuchlinesgeneratedinequalpartsoftime。
31。Apointmaybethelimitofaline：alinemaybethelimitofasurface：amomentmayterminatetime。Buthowcanweconceiveavelocitybythehelpofsuchlimits？Itnecessarilyimpliesbothtimeandspace，andcannotbeconceivedwithoutthem。Andifthevelocitiesofnascentandevanescentquantities，i。e。abstractedfromtimeandspace，maynotbecomprehended，howcanwecomprehendanddemonstratetheirproportions；orconsidertheirrationesprimaeandultimae？