OftheotherQualitiesofourIdeaofSpaceandTime。
Nodiscoverycou’dhavebeenmademorehappilyfordecidingallcontroversiesconcerningideas，thanthatabovemention’d，thatimpressionsalwaystaketheprecedencyofthem，andthateveryidea，withwhichtheimaginationisfurnish’d，firstmakesitsappearanceinacorrespondentimpression。Theselatterperceptionsareallsoclearandevident，thattheyadmitofnocontroversy；tho’manyofourideasaresoobscure，that’tisalmostimpossibleevenforthemind，whichformsthem，totellexactlytheirnatureandcomposition。Letusapplythisprinciple，inordertodiscoverfartherthenatureofourideasofspaceandtime。
Uponopeningmyeyes，andturningthemtothesurroundingobjects，Iperceivemanyvisiblebodies；anduponshuttingthemagain，andconsideringthedistancebetwixtthesebodies，Iacquiretheideaofextension。Aseveryideaisderiv’dfromsomeimpression，whichisexactlysimilartoit，theimpressionssimilartothisideaofextension，musteitherbesomesensationsderiv’dfromthesight，orsomeinternalimpressionsarisingfromthesesensations。，Ourinternalimpressionsareourpassions，emotions，desiresandaversions；noneofwhich，Ibelieve，willeverbeassertedtobethemodel，fromwhichtheideaofspaceisderiv’d。Thereremainsthereforenothingbutthesenses，whichcanconveytousthisoriginalimpression。Nowwhatimpressiondooarsenseshereconveytous？Thisistheprincipalquestion，anddecideswithoutappealconcerningthenatureoftheidea。
Thetablebeforemeisalonesufficientbyitsviewtogivemetheideaofextension。Thisidea，then，isborrow’dfrom，andrepresentssomeimpression，whichthismomentappearstothesenses。Butmysensesconveytomeonlytheimpressionsofcolour’dpoints，dispos’dina，certainmanner。Iftheeyeissensibleofanythingfarther，Idesireitmaybepointedouttome。Butifitbeimpossibletoshewanythingfarther，wemayconcludewithcertainty，thattheideaofextensionisnothingbutacopyofthesecolour’dpoints，andofthemanneroftheirappearance。
Supposethatintheextendedobject，orcompositionofcolour’dpoints，fromwhichwefirstreceiv’dtheideaofextension，thepointswereofapurplecolour；itfollows，thatineveryrepetitionofthatideawewou’dnotonlyplacethepointsinthesameorderwithrespecttoeachother，butalsobestowonthemthatprecisecolour，withwhichaloneweareacquainted。Butafterwardshavingexperienceoftheothercoloursofviolet，green，red，white，black，andofallthedifferentcompositionsofthese，andfindingaresemblanceinthedispositionofcolour’dpoints，ofwhichtheyarecompos’d，weomitthepeculiaritiesofcolour，asfaraspossible，andfoundanabstractideamerelyonthatdispositionofpoints，ormannerofappearance，inwhichtheyagree。Nayevenwhentheresemblanceiscarry’dbeyondtheobjectsofonesense，andtheimpressionsoftoucharefoundtobeSimilartothoseofsightinthedispositionoftheirparts；thisdoesnothindertheabstractideafromrepresentingboth，uponaccountoftheirresemblance。Allabstractideasarereallynothingbutparticularones，consider’dinacertainlight；butbeingannexedtogeneralterms，theyareabletorepresentavastvariety，andtocomprehendobjects，which，astheyarealikeinsomeparticulars，areinothersvastlywideofeachother。’
Theideaoftime，beingderiv’dfromthesuccessionofourperceptionsofeverykind，ideasaswellasimpressions，andimpressionsofreflectionaswellasofsensationswillaffordusaninstanceofanabstractidea，whichcomprehendsastillgreatervarietythanthatofspace，andyetisrepresentedinthefancybysomeparticularindividualideaofadeterminatequantityandquality。
As’tisfromthedispositionofvisibleandtangibleobjectswereceivetheideaofspace，sofromthesuccessionofideasandimpressionsweformtheideaoftime，norisitpossiblefortimealoneevertomakeitsappearance，orbetakennoticeofbythemind。Amaninasoundsleep，orstronglyoccupy’dwithonethought，isinsensibleoftime；andaccordingashisperceptionssucceedeachotherwithgreaterorlessrapidity，thesamedurationappearslongerorshortertohisimagination。Ithasbeenremark’dbyagreatphilosopher，(7)thatourperceptionshavecertainboundsinthisparticular，whicharefix’dbytheoriginalnatureandconstitutionofthemind，andbeyondwhichnoinfluenceofexternalobjectsonthesensesiseverabletohastenorretardourthought。Ifyouwheelaboutaburningcoalwithrapidity，itwillpresenttothesensesanimageofacircleoffire；norwillthereseemtobeanyintervaloftimebetwixtitsrevolutions；meerlybecause’tisimpossibleforourperceptionstosucceedeachotherwiththesamerapidity，thatmotionmaybecommunicatedtoexternalobjects。Whereverwehavenosuccessiveperceptions，wehavenonotionoftime，eventho’therebearealsuccessionintheobjects。Fromthesephenomena，aswellasfrommanyothers，wemayconclude，thattimecannotmakeitsappearancetothemind，eitheralone，orattendedwithasteadyunchangeableobject，butisalwaysdiscoveredsomeperceivablesuccessionofchangeableobjects。
Toconfirmthiswemayaddthefollowingargument，whichtomeseemsperfectlydecisiveandconvincing。’Tisevident，thattimeordurationconsistsofdifferentparts:Forotherwisewecou’dnotconceivealongerorshorterduration。’Tisalsoevident，thatthesepartsarenotco-existent:Forthatqualityoftheco-existenceofpartsbelongstoextension，andiswhatdistinguishesitfromduration。Nowastimeiscompos’dofparts，thatarenotcoexistent:anunchangeableobject，sinceitproducesnonebutcoexistentimpressions，producesnonethatcangiveustheideaoftime；andconsequentlythatideamustbederiv’dfromasuccessionofchangeableobjects，andtimeinitsfirstappearancecanneverbesever’dfromsuchasuccession。
Havingthereforefound，thattimeinitsfirstappearancetothemindisalwaysconjoin’dwithasuccessionofchangeableobjects，andthatotherwiseitcanneverfallunderournotice，wemustnowexaminewhetheritcanbeconceiv’dwithoutourconceivinganysuccessionofobjects，andwhetheritcanaloneformadistinctideaintheimagination。
Inordertoknowwhetheranyobjects，whicharejoin’dinimpression，beinseparableinidea，weneedonlyconsider，iftheybedifferentfromeachother；inwhichcase，’tisplaintheymaybeconceiv’dapart。Everything，thatisdifferentisdistinguishable:andeverything，thatisdistinguishable，maybeseparated，accordingtothemaximsabove-explain’d。Ifonthecontrarytheybenotdifferent，theyarenotdistinguishable:andiftheybenotdistinguishable，theycannotbeseparated。Butthisispreciselythecasewithrespecttotime，compar’dwithoursuccessiveperceptions。Theideaoftimeisnotderiv’dfromaparticularimpressionmix’dupwithothers，andplainlydistinguishablefromthem；butarisesaltogetherfromthemanner，inwhichimpressionsappeartothemind，withoutmakingoneofthenumber。Fivenotesplay’donaflutegiveustheimpressionandideaoftime；tho’timebenotasixthimpression，whichpresentsitselftothehearingoranyotherofthesenses。Norisitasixthimpression，whichthemindbyreflectionfindsinitself。Thesefivesoundsmakingtheirappearanceinthisparticularmanner，excitenoemotioninthemind，norproduceanaffectionofanykind，whichbeingobserv’dbyitcangiverisetoanewidea。Forthatisnecessarytoproduceanewideaofreflection，norcanthemind，byrevolvingoverathousandtimesallitsideasofsensation，everextractfromthemanyneworiginalidea，unlessnaturehassofram’ditsfaculties，thatitfeelssomeneworiginalimpressionarisefromsuchacontemplation。Buthereitonlytakesnoticeofthemanner，inwhichthedifferentsoundsmaketheirappearance；andthatitmayafterwardsconsiderwithoutconsideringtheseparticularsounds，butmayconjoinitwithanyotherobjects。Theideasofsomeobjectsitcertainlymusthave，norisitpossibleforitwithouttheseideasevertoarriveatanyconceptionoftime；whichsinceit，appearsnotasanyprimarydistinctimpression，canplainlybenothingbutdifferentideas，orimpressions，orobjectsdispos’dinacertainmanner，thatis，succeedingeachother。
Iknowtherearesomewhopretend，thattheideaofdurationisapplicableinapropersensetoobjects，whichareperfectlyunchangeable；andthisItaketobethecommonopinionofphilosophersaswellasofthevulgar。Buttobeconvinc’dofitsfalsehoodweneedbutreflectontheforegoingconclusion，thattheideaofdurationisalwaysderiv’dfromasuccessionofchangeableobjects，andcanneverbeconvey’dtothemindbyanythingstedfastandunchangeable。Foritinevitablyfollowsfromthence，thatsincetheideaofdurationcannotbederiv’dfromsuchanobject，itcannever-inanyproprietyorexactnessbeapply’dtoit，norcananythingunchangeablebeeversaidtohaveduration。IdeasalwaysrepresenttheObjectsorimpressions，fromwhichtheyarederiv’d，andcanneverwithoutafictionrepresentorbeapply’dtoanyother。Bywhatfictionweapplytheideaoftime，eventowhatisunchangeable，andsuppose，asiscommon，thatdurationisameasureofrestaswellasofmotion，weshallconsider(8)afterwards。
Thereisanotherverydecisiveargument，whichestablishesthepresentdoctrineconcerningourideasofspaceandtime，andisfoundedonlyonthatsimpleprinciple，thatourideasofthemarecompoundedofparts，whichareindivisible。Thisargumentmaybeworththeexamining。
Everyidea，thatisdistinguishable，beingalsoseparable，letustakeoneofthosesimpleindivisibleideas，ofwhichthecompoundoneofextensionisform’d，andseparatingitfromallothers，andconsideringitapart，letusformajudgmentofitsnatureandqualities。
’Tisplainitisnottheideaofextension。Fortheideaofextensionconsistsofparts；andthisidea，accordingtot-hesupposition，isperfectlysimpleandindivisible。Isitthereforenothing？Thatisabsolutelyimpossible。Forasthecompoundideaofextension，whichisreal，iscompos’dofsuchideas；werethesesomanynon-entities，therewou’dbearealexistencecompos’dofnon-entities；whichisabsurd。HerethereforeImustask，Whatisourideaofasimpleandindivisiblepoint？Nowonderifmyanswerappearsomewhatnew，sincethequestionitselfhasscarceeveryetbeenthoughtof。Wearewonttodisputeconcerningthenatureofmathematicalpoints，butseldomconcerningthenatureoftheirideas。
Theideaofspaceisconvey’dtothe。mindbytwosenses，thesightandtouch；nordoesanythingeverappearextended，thatisnoteithervisibleortangible。Thatcompoundimpression，whichrepresentsextension，consistsofseverallesserimpressions，thatareindivisibletotheeyeorfeeling，andmaybecall’dimpressionsofatomsorcorpusclesendow’dwithcolourandsolidity。Butthisisnotall。’Tisnotonlyrequisite，thattheseatomsshou’dbecolour’dortangible，inordertodiscoverthemselvestooursenses；’tisalsonecessaryweshou’dpreservetheideaoftheircolourortangibilityinordertocomprehendthembyourimagination。Thereisnothingbuttheideaoftheircolourortangibility，whichcanrenderthemconceivablebythemind。Upontheremovaloftheideasofthesesensiblequalities，theyareutterlyannihilatedtothethoughtorimagination。’
Nowsuchasthepartsare，suchisthewhole。Ifapointbenotconsider’dascolour’dortangible，itcanconveytousnoidea；andconsequentlytheideaofextension，whichiscompos’doftheideasofthesepoints，canneverpossiblyexist。Butiftheideaofextensionreallycanexist，asweareconsciousitdoes，itspartsmustalsoexist；andinordertothat，mustbeconsider’dascolour’dortangible。Wehavethereforenoideaofspaceorextension，butwhenweregarditasanobjecteitherofoursightorfeeling。
Thesamereasoningwillprove，thattheindivisiblemomentsoftimemustbefill’dwithsomerealobjectorexistence，whosesuccessionformstheduration，andmakesitbeconceivablebythemind。
Objectionsanswer’d。
Oursystemconcerningspaceandtimeconsistsoftwoparts，whichareintimatelyconnectedtogether。Thefirstdependsonthischainofreasoning。Thecapacityofthemindisnotinfinite；consequentlynoideaofextensionordurationconsistsofaninfinitenumberofpartsorinferiorideas，butofafinitenumber，andthesesimpleandindivisible:’Tisthereforepossibleforspaceandtimetoexistconformabletothisidea:Andifitbepossible，’tiscertaintheyactuallydoexistconformabletoit；sincetheirinfinitedivisibilityisutterlyimpossibleandcontradictory。
Theotherpartofoursystemisaconsequenceofthis。Theparts，intowhichtheideasofspaceandtimeresolvethemselves，becomeatlastindivisible；andtheseindivisibleparts，beingnothinginthemselves，areinconceivablewhennotfill’dwithsomethingrealandexistent。Theideasofspaceandtimearethereforenoseparateordistinctideas，butmerelythoseofthemannerororder，inwhichobjectsexist:Orinotherwords，’tisimpossibletoconceiveeitheravacuumandextensionwithoutmatter，oratime，whentherewasnosuccessionorchangeinanyrealexistence。Theintimateconnexionbetwixtthesepartsofoursystemisthereasonwhyweshallexaminetogethertheobjections，whichhavebeenurg’dagainstbothofthem，beginningwiththoseagainstthefinitedivisibilityofextension。
I。Thefirstoftheseobjections，whichIshalltakenoticeof，ismorepropertoprovethisconnexionanddependenceoftheonepartupontheother，thantodestroyeitherofthem。Ithasoftenbeenmaintainedintheschools，thatextensionmustbedivisible，ininfinitum，becausethesystemofmathematicalpointsisabsurd；andthatsystemisabsurd，becauseamathematicalpointisanon-entity，andconsequentlycanneverbyitsconjunctionwithothersformarealexistence。Thiswou’dbeperfectlydecisive，weretherenomediumbetwixttheinfinitedivisibilityofmatter，andthenon-entityofmathematicalpoints。Butthereisevidentlyamedium，viz。thebestowingacolourorsolidityonthesepoints；andtheabsurdityofboththeextremesisademonstrationofthetruthandrealityofthismedium。Thesystemofphysicalpoints，whichisanothermedium，istooabsurdtoneedarefutation。Arealextension，suchasaphysicalpointissuppos’dtobe，canneverexistwithoutparts，differentfromeachother；andwhereverobjectsaredifferent，theyaredistinguishableandseparablebytheimagination。
II。Thesecondobjectionisderiv’dfromthenecessitytherewou’dbeofpenetration，ifextensionconsistedofmathematicalpoints。Asimpleandindivisibleatom，thattouchesanother，mustnecessarilypenetrateit；for’tisimpossibleitcantouchitbyitsexternalparts，fromtheverysuppositionofitsperfectsimplicity，whichexcludesallparts。Itmustthereforetouchitintimately，andinitswholeessence，secundumse，tota，&；totaliter；whichistheverydefinitionofpenetration。Butpenetrationisimpossible:Mathematicalpointsareofconsequenceequallyimpossible。
Ianswerthisobjectionbysubstitutingajusterideaofpenetration。Supposetwobodiescontainingnovoidwithintheircircumference，toapproacheachother，andtouniteinsuchamannerthatthebody，whichresultsfromtheirunion，isnomoreextendedthaneitherofthem；’tisthiswemustmeanwhenwetalkofpenetration。But’tisevidentthispenetrationisnothingbuttheannihilationofoneofthesebodies，andthepreservationoftheother，withoutourbeingabletodistinguishparticularlywhichispreserv’dandwhichannihilated。Beforetheapproachwehavetheideaoftwobodies。Afteritwehavetheideaonlyofone。’Tisimpossibleforthemindtopreserveanynotionofdifferencebetwixttwobodiesofthesamenatureexistinginthesameplaceatthesametime。
Takingthenpenetrationinthissense，fortheannihilationofonebodyuponitsapproachtoanother，Iaskanyone，ifheseesanecessity，thatacolour’dortangiblepointshou’dbeannihilatedupontheapproachofanothercolour’dortangiblepoint？Onthecontrary，doeshenotevidentlyperceive，thatfromtheunionofthesepointsthereresultsanobject，whichiscompoundedanddivisible，andmaybedistinguishedintotwoparts，ofwhicheachpreservesitsexistencedistinctandseparate，notwithstandingitscontiguitytotheother？Lethimaidhisfancybyconceivingthesepointstobeofdifferentcolours，thebettertopreventtheircoalitionandconfusion。Ablueandaredpointmaysurelyliecontiguouswithoutanypenetrationorannihilation。Foriftheycannot，whatpossiblycanbecomeofthem？Whethershalltheredorthebluebeannihilated？Orifthesecoloursuniteintoone，whatnewcolourwilltheyproducebytheirunion？
Whatchieflygivesrisetotheseobjections，andatthesametimerendersitsodifficulttogiveasatisfactoryanswertothem，isthenaturalinfirmityandunsteadinessbothofourimaginationandsenses，whenemploy’donsuchminuteobjects。Putaspotofinkuponpaper，andretiretosuchadistance，thatthespotbecomesaltogetherinvisible；youwillfind，thatuponyourreturnandnearerapproachthespotfirstbecomesvisiblebyshortintervals；andafterwardsbecomesalwaysvisible；andafterwardsacquiresonlyanewforceinitscolouringwithoutaugmentingitsbulk；andafterwards，whenithasencreas’dtosuchadegreeastobereallyextended，’tisstilldifficultfortheimaginationtobreakitintoitscomponentparts，becauseoftheuneasinessitfindsintheconceptionofsuchaminuteobjectasasinglepoint。Thisinfirmityaffectsmostofourreasoningsonthepresentsubject，andmakesitalmostimpossibletoanswerinanintelligiblemanner，andinproperexpressions，manyquestionswhichmayariseconcerningit。
III。Therehavebeenmanyobjectionsdrawnfromthemathematicsagainsttheindivisibilityofthepartsofextension:tho’atfirstsightthatscienceseemsratherfavourabletothepresentdoctrine；andifitbecontraryinitsdemonstrations，’tisperfectlyconformableinitsdefinitions。Mypresentbusinessthenmustbetodefendthedefinitions，andrefutethedemonstrations。
Asurfaceisdefin’dtobelengthandbreadthwithoutdepth:Alinetobelengthwithoutbreadthordepth:Apointtobewhathasneitherlength，breadthnordepth。’Tisevidentthatallthisisperfectlyunintelligibleuponanyothersuppositionthanthatofthe。compositionofextensionbyindivisiblepointsoratoms。Howelsecou’danythingexistwithoutlength，withoutbreadth，orwithoutdepth？
Twodifferentanswers，Ifind，havebeenmadetothisargument；neitherofwhichisinmyopinionsatisfactory。Thefirstis，thattheobjectsofgeometry，thosesurfaces，linesandpoints，whoseproportionsandpositionsitexamines，aremereideasinthemind；Iandnotonlyneverdid，butnevercanexistinnature。Theyneverdidexist；fornoonewillpretendtodrawalineormakeasurfaceentirelyconformabletothedefinition:Theynevercanexist；forwemayproducedemonstrationsfromtheseveryideastoprove，thattheyareimpossible。
Butcananythingbeimagin’dmoreabsurdandcontradictorythanthisreasoning？Whatevercanbeconceiv’dbyaclearanddistinctideanecessarilyimpliesthepossibilityofexistence；andhewhopretendstoprovetheimpossibilityofitsexistencebyanyargumentderivedfromtheclearidea，inrealityasserts，thatwehavenoclearideaofit，becausewehaveaclearidea。’Tisinvaintosearchforacontradictioninanythingthatisdistinctlyconceiv’dbythemind。Diditimplyanycontradiction，’tisimpossibleitcou’deverbeconceiv’d。
Thereisthereforenomediumbetwixtallowingatleastthepossibilityofindivisiblepoints，anddenyingtheiridea；and’tisonthislatterprinciple，thatthesecondanswertotheforegoingargumentisfounded。Ithasbeen(9)pretended，thattho’itbeimpossibletoconceivealengthwithoutanybreadth，yetbyanabstractionwithoutaseparation，wecanconsidertheonewithoutregardingtheother；inthesamemanneraswemaythinkofthelengthofthewaybetwixttwotowns，andoverlookitsbreadth。Thelengthisinseparablefromthebreadthbothinnatureandinourminds；butthisexcludesnotapartialconsideration，andadistinctionofreason，afterthemanneraboveexplain’d。
InrefutingthisanswerIshallnotinsistontheargument，whichIhavealreadysufficientlyexplained，thatifitbeimpossibleforthemindtoarriveataminimuminitsideas，itscapacitymustbeinfinite，inordertocomprehendtheinfinitenumberofparts，ofwhichitsideaofanyextensionwou’dbecompos’d。Ishallhereendeavourtofindsomenewabsurditiesinthisreasoning。
Asurfaceterminatesasolid；alineterminatesasurface；apointterminatesaline；butIassert，thatiftheideasofapoint，lineorsurfacewerenotindivisible，’tisimpossibleweshou’deverconceivetheseterminations:Forlettheseideasbesuppos’dinfinitelydivisible；andthenletthefancyendeavourtofixitselfontheideaofthelastsurface，lineorpoint；itimmediatelyfindsthisideatobreakintoparts；anduponitsseizingthelastoftheseparts，itlosesitsholdbyanewdivision，andsoonininfinitum，withoutanypossibilityofitsarrivingataconcludingidea。Thenumberoffractionsbringitnonearerthelastdivision，thanthefirstideaitform’d。Everyparticleeludesthegraspbyanewfraction；likequicksilver，whenweendeavourtoseizeit。Butasinfacttheremustbesomething，whichterminatestheideaofeveryfinitequantity；andasthisterminatingideacannotitselfconsistofpartsorinferiorideas；otherwiseitwou’dbethelastofitsparts，whichfinish’dtheidea，andsoon；thisisaclearproof，thattheideasofsurfaces，linesandpointsadmitnotofanydivision；thoseofsurfacesindepth；oflinesinbreadthanddepth；andofpointsinanydimension。
Theschoolweresosensibleoftheforceofthisargument，thatsomeofthemmaintained，thatnaturehasmix’damongthoseparticlesofmatter，whicharedivisibleininfinitum，anumberofmathematicalpoints，inordertogiveaterminationtobodies；andotherseludedtheforceofthisreasoningbyaheapofunintelligiblecavilsanddistinctions。Boththeseadversariesequallyyieldthevictory。Amanwhohideshimself，confessesasevidentlythesuperiorityofhisenemy，asanother，whofairlydelivershisarms。
Thusitappears，thatthedefinitionsofmathematicsdestroythepretendeddemonstrations；andthatifwehavetheideaofindivisiblepoints，linesandsurfacesconformabletothedefinition，theirexistenceiscertainlypossible:butifwehavenosuchidea，’tisimpossiblewecaneverconceivetheterminationofanyfigure；withoutwhichconceptiontherecanbenogeometricaldemonstration。
ButIgofarther，andmaintain，thatnoneofthesedemonstrationscanhavesufficientweighttoestablishsuchaprinciple，asthisofinfinitedivisibility；andthatbecausewithregardtosuchminuteobjects，theyarenotproperlydemonstrations，beingbuiltonideas，whicharenotexact，andmaxims，whicharenotpreciselytrue。Whengeometrydecidesanythingconcerningtheproportionsofquantity，weoughtnottolookfortheutmostprecisionandexactness。Noneofitsproofsextendsofar。Ittakesthedimensionsandproportionsoffiguresjustly；butroughly，andwithsomeliberty。Itserrorsareneverconsiderable；norwou’diterratall，diditnotaspiretosuchanabsoluteperfection。
Ifirstaskmathematicians，whattheymeanwhentheysayonelineorsurfaceisEQUALto，orGREATERorLESSthananother？Letanyofthemgiveananswer，towhateversecthebelongs，andwhetherhemaintainsthecompositionofextensionbyindivisiblepoints，orbyquantitiesdivisibleininfinitum。Thisquestionwillembarrassbothofthem。
Therearefewornomathematicians，whodefendthehypothesisofindivisiblepoints；andyetthesehavethereadiestandjustestanswertothepresentquestion。Theyneedonlyreply，thatlinesorsurfacesareequal，whenthenumbersofpointsineachareequal；andthatastheproportionofthenumbersvaries，theproportionofthelinesandsurfacesisalsovary’d。Buttho’thisanswerbejust，aswellasobvious；yetImayaffirm，thatthisstandardofequalityisentirelyuseless，andthatitneverisfromsuchacomparisonwedetermineobjectstobeequalorunequalwithrespecttoeachother。Forasthepoints，whichenterintothecompositionofanylineorsurface，whetherperceiv’dbythesightortouch，aresominuteandsoconfoundedwitheachother，that’tisutterlyimpossibleforthemindtocomputetheirnumber，suchacomputationwillNeveraffordusastandardbywhichwemayjudgeofproportions。Noonewilleverbeabletodeterminebyanexactnumeration，thataninchhasfewerpointsthanafoot，orafootfewerthananelloranygreatermeasure:forwhichreasonweseldomorneverconsiderthisasthestandardofequalityorinequality。
Astothose，whoimagine，thatextensionisdivisibleininfinitum，’tisimpossibletheycanmakeuseofthisanswer，orfixtheequalityofanylineorsurfacebyanumerationofitscomponentparts。Forsince，accordingtotheirhypothesis，theleastaswellasgreatestfigurescontainaninfinitenumberofparts；andsinceinfinitenumbers，properlyspeaking，canneitherbeequalnorunequalwithrespecttoeachother；theequalityorinequalityofanyportionsofspacecanneverdependonanyproportioninthenumberoftheirparts。’Tistrue，itmaybesaid，thattheinequalityofanellandayardconsistsinthedifferentnumbersofthefeet，ofwhichtheyarecompos’d；andthatofafootandayardinthenumberoftheinches。Batasthatquantitywecallaninchintheoneissuppos’dequaltowhatwecallaninchintheother，andas’tisimpossibleforthemindtofindthisequalitybyproceedingininfinitumwiththesereferencestoinferiorquantities:’tisevident，thatatlastwemustfixsomestandardofequalitydifferentfromanenumerationoftheparts。
Therearesome，(10)whopretend，thatequalityisbestdefin’dbycongruity，andthatanytwofiguresareequal，whenupontheplacingofoneupontheother，alltheirpartscorrespondtoandtoucheachother。Inordertojudgeofthisdefinitionletusconsider，thatsinceequalityisarelation，itisnot，strictlyspeaking，apropertyinthefiguresthemselves，butarisesmerelyfromthecomparison，whichthemindmakesbetwixtthem。’Ifitconsists，therefore，inthisimaginaryapplicationandmutualcontactofparts，wemustatleasthaveadistinctnotionoftheseparts，andmustconceivetheircontact。Now’tisplain，thatinthisconceptionwewou’drunupthesepartstothegreatestminuteness，whichcanpossiblybeconceiv’d；sincethecontactoflargepartswou’dneverrenderthefiguresequal。Buttheminutestpartswecanconceivearemathematicalpoints；andconsequentlythisstandardofequalityisthesamewiththatderiv’dfromtheequalityofthenumberofpoints；whichwehavealreadydeterminedtobeajustbutanuselessstandard。Wemustthereforelooktosomeotherquarterforasolutionofthepresentdifficulty。
[ThefollowingparagraphisaddedfromtheappendixtoBookIII]
Therearemanyphilosophers，whorefusetoassignanystandardofequality，butassert，that’tissufficienttopresenttwoobjects，thatareequal，inordertogiveusajustnotionofthisproportion。Alldefinitions，saythey，arefruitless，withouttheperceptionofsuchobjects；andwhereweperceivesuchobjects，wenolongerstandinneedofanydefinition。Tothisreasoning，Ientirelyagree。；andassert，thattheonlyusefulnotionofequality，orinequality，isderiv’dfromthewholeunitedappearanceandthecomparisonofparticularobjects。
’Tisevident，thattheeye，orratherthemindisoftenableatoneviewtodeterminetheproportionsofbodies，andpronouncethemequalto，orgreaterorlessthaneachother，withoutexaminingorcomparingthenumberoftheirminuteparts。Suchjudgmentsarenotonlycommon，butinmanycasescertainandinfallible。Whenthemeasureofayardandthatofafootarepresented，themindcannomorequestion，thatthefirstislongerthanthesecond，thanitcandoubtofthoseprinciples，whicharethemostclearandself-evident。
Therearethereforethreeproportions，whichtheminddistinguishesinthegeneralappearanceofitsobjects，andcallsbythenamesofgreater，lessandequal。Buttho’itsdecisionsconcerningtheseproportionsbesometimesinfallible，theyarenotalwaysso；norareourjudgmentsofthiskindmoreexemptfromdoubtanderrorthanthoseonanyothersubject。Wefrequentlycorrectourfirstopinionbyareviewandreflection；andpronouncethoseobjectstobeequal，whichatfirstweesteem’dunequal；andregardanobjectasless，tho’beforeitappear’dgreaterthananother。Noristhistheonlycorrection，whichthesejudgmentsofoursensesundergo；butweoftendiscoverourerrorbyajuxtapositionoftheobjects；orwherethatisimpracticable，bytheuseofsomecommonandinvariablemeasure，whichbeingsuccessivelyapply’dtoeach，informsusoftheirdifferentproportions。Andeventhiscorrectionissusceptibleofanewcorrection。，andofdifferentdegreesofexactness，accordingtothenatureoftheinstrument，bywhichwemeasurethebodies，andthecarewhichweemployinthecomparison。’
Whenthereforethemindisaccustomedtothesejudgmentsandtheircorrections，andfindsthatthesameproportionwhichmakestwofigureshaveintheeyethatappearance，whichwecallequality，makesthemalsocorrespondtoeachother，andtoanycommonmeasure，withwhichtheyarecompar’d，weformamix’dnotionofequalityderiv’dbothfromthelooserandstrictermethodsofcomparison。Butwearenotcontentwiththis。Forassoundreasonconvincesusthattherearebodiesvastlymoreminutethanthose，whichappeartothesenses；andasafalsereasonwou’dperswadeus，thattherearebodiesinfinitelymoreminute；weclearlyperceive，thatwearenotpossess’dofanyinstrumentorartofmeasuring，whichcansecureusfromillerroranduncertainty。Wearesensible，thattheadditionorremovalofoneoftheseminuteparts，isnotdiscernibleeitherintheappearanceormeasuring；andasweimagine，thattwofigures，whichwereequalbefore，cannotbeequalafterthisremovaloraddition，wethereforesupposesomeimaginarystandardofequality，bywhichtheappearancesandmeasuringareexactlycorrected，andthefiguresreduc’dentirelytothatproportion。Thisstandardisplainlyimaginary。Forastheveryideaofequalityisthatofsuchaparticularappearancecorrectedbyjuxtapositionoracommonmeasure。thenotionofanycorrectionbeyondwhatwehaveinstrumentsandarttomake，isamerefictionofthemind，anduselessaswellasincomprehensible。Buttho’thisstandardbeonlyimaginary，thefictionhoweverisverynatural；norisanythingmoreusual，thanforthemindtoproceedafterthismannerwithanyaction，evenafterthereasonhasceas’d，whichfirstdeterminedittobegin。Thisappearsveryconspicuouslywithregardtotime；wheretho’’tisevidentwehavenoexactmethodofdeterminingtheproportionsofparts，notevensoexactasinextension，yetthevariouscorrectionsofourmeasures，andtheirdifferentdegreesofexactness，havegivenasanobscureandimplicitnotionofaperfectandentireequality。Thecaseisthesameinmanyothersubjects。Amusicianfindinghisearbecomingeverydaymoredelicate，andcorrectinghimselfbyreflectionandattention，proceedswiththesameactofthemind，evenwhenthesubjectfailshim，andentertainsanotionofacompleattierceoroctave，withoutbeingabletotellwhencehederiveshisstandard。Apainterformsthesamefictionwithregardtocolours。Amechanicwithregardtomotion。Totheonelightandshade；totheotherswiftandslowareimagin’dtobecapableofanexactcomparisonandequalitybeyondthejudgmentsofthesenses。
WemayapplythesamereasoningtoCURVEandRIGHTlines。Nothingismoreapparenttothesenses，thanthedistinctionbetwixtacurveandarightline；norarethereanyideaswemoreeasilyformthantheideasoftheseobjects。Buthowevereasilywemayformtheseideas，’tisimpossibletoproduceanydefinitionofthem，whichwillfixthepreciseboundariesbetwixtthem。Whenwedrawlinesuponpaper，oranycontinu’dsurface，thereisacertainorder，bywhichthelinesrunalongfromonepointtoanother，thattheymayproducetheentireimpressionofacurveorrightline；butthisorderisperfectlyunknown，andnothingisobserv’dbuttheunitedappearance。Thusevenuponthesystemofindivisiblepoints，wecanonlyformadistantnotionofsomeunknownstandardtotheseobjects。Uponthatofinfinitedivisibilitywecannotgoeventhislength；butarereduc’dmeerlytothegeneralappearance，astherulebywhichwedeterminelinestobeeithercurveorrightones。Buttho’wecangivenoperfectdefinitionoftheselines，norproduceanyveryexactmethodofdistinguishingtheonefromtheother；yetthishindersusnotfromcorrectingthefirstappearancebyamoreaccurateconsideration，andbyacomparisonwithsomerule，ofwhoserectitudefromrepeatedtrialswehaveagreaterassurance。And’tisfromthesecorrections，andbycarryingonthesameactionofthemind，evenwhenitsreasonfailsus，thatweformthelooseideaofaperfectstandardtothesefigures，withoutbeingabletoexplainorcomprehendit。
’Tistrue，mathematicianspretendtheygiveanexactdefinitionofarightline，whentheysay，itistheshortestwaybetwixttwopoints。ButinthefirstplaceIobserve，thatthisismoreproperlythediscoveryofoneofthepropertiesofarightline，thanajustdeflationofit。ForIaskanyone，ifuponmentionofarightlinehethinksnotimmediatelyonsuchaparticularappearance，andif’tisnotbyaccidentonlythatheconsidersthisproperty？Arightlinecanbecomprehendedalone；butthisdefinitionisunintelligiblewithoutacomparisonwithotherlines，whichweconceivetobemoreextended。Incommonlife’tisestablishedasamaxim，thatthestraightestwayisalwaystheshortest；whichwou’dbeasabsurdastosay，theshortestwayisalwaystheshortest，ifourideaofarightlinewasnotdifferentfromthatoftheshortestwaybetwixttwopoints。
Secondly，IrepeatwhatIhavealreadyestablished，thatwehavenopreciseideaofequalityandinequality，shorterandlonger，morethanofarightlineoracurve；andconsequentlythattheonecanneveraffordusaperfectstandardfortheother。Anexactideacanneverbebuiltonsuchasarelooseandundetermined。
Theideaofaplainsurfaceisaslittlesusceptibleofaprecisestandardasthatofarightline；norhaveweanyothermeansofdistinguishingsuchasurface，thanitsgeneralappearance。’Tisinvain，thatmathematiciansrepresentaplainsurfaceasproduc’dbytheflowingofarightline。’Twillimmediatelybeobjected，thatourideaofasurfaceisasindependentofthismethodofformingasurface，asourideaofanellipseisofthatofacone；thattheideaofarightlineisnomoreprecisethanthatofaplainsurface；thatarightlinemayflowirregularly，andbythatmeansformafigurequitedifferentfromaplane；andthatthereforewemustsupposeittoflowalongtworightlines，paralleltoeachother，andonthesameplane；whichisadescription，thatexplainsathingbyitself，andreturnsinacircle。
Itappears，then，thattheideaswhicharemostessentialtogeometry，viz。thoseofequalityandinequality，ofarightlineandaplainsurface，arefarfrombeingexactanddeterminate，accordingtoourcommonmethodofconceivingthem。Notonlyweareincapableoftelling，ifthecasebeinanydegreedoubtful，whensuchparticularfiguresareequal；whensuchalineisarightone，andsuchasurfaceaplainone；butwecanformnoideaofthatproportion，orofthesefigures，whichisfirmandinvariable。Ourappealisstilltotheweakandfalliblejudgment，whichwemakefromtheappearanceoftheobjects，andcorrectbyacompassorcommonmeasure；andifwejointhesuppositionofanyfarthercorrection，’tisofsuch-a-oneasiseitheruselessorimaginary。Invainshou’dwehaverecoursetothecommontopic，andemploythesuppositionofadeity，whoseomnipotencemayenablehimtoformaperfectgeometricalfigure，anddescribearightlinewithoutanycurveorinflexion。Astheultimatestandardofthesefiguresisderiv’dfromnothingbutthesensesandimagination，’tisabsurdtotalkofanyperfectionbeyondwhatthesefacultiescanjudgeof；sincethetrueperfectionofanythingconsistsinitsconformitytoitsstandard。
Nowsincetheseideasaresolooseanduncertain，Iwou’dfainaskanymathematicianwhatinfallibleassurancehehas，notonlyofthemoreintricate，andobscurepropositionsofhisscience，butofthemostvulgarandobviousprinciples？Howcanheprovetome，forinstance，thattworightlinescannothaveonecommonsegment？Orthat’tisimpossibletodrawmorethanonerightlinebetwixtanytwopoints？Shou’dbetellme，thattheseopinionsareobviouslyabsurd，andrepugnanttoourclearideas；Iwouldanswer，thatIdonotdeny，wheretworightlinesinclineuponeachotherwithasensibleangle，but’tisabsurdtoimaginethemtohaveacommonsegment。Butsupposingthesetwolinestoapproachattherateofaninchintwentyleagues，Iperceivenoabsurdityinasserting，thatupontheircontacttheybecomeone。For，Ibeseechyou，bywhatruleorstandarddoyoujudge，whenyouassert，thattheline，inwhichIhavesuppos’dthemtoconcur，cannotmakethesamerightlinewiththosetwo，thatformsosmallananglebetwixtthem？Youmustsurelyhavesomeideaofarightline，towhichthislinedoesnotagree。Doyouthereforemeanthatittakesnotthepointsinthesameorderandbythesamerule。，asispeculiarandessentialtoarightline？Ifso，Imustinformyou，thatbesidesthatinjudgingafterthismanneryouallow，thatextensioniscompos’dofindivisiblepoints(which，perhaps，ismorethanyouintend)besidesthis，Isay，Imustinformyou，thatneitheristhisthestandardfromwhichweformtheideaofarightline；nor，ifitwere，isthereanysuchfirmnessinour-sensesorimagination，astodeterminewhensuchanorderisviolatedorpreserv’d。Theoriginalstandardofarightlineisinrealitynothingbutacertaingeneralappearance；and’tisevidentrightlinesmaybemadetoconcurwitheachother，andyetcorrespondtothisstandard，tho’correctedbyallthemeanseitherpracticableorimaginable。
[ThisparagraphisinsertedfromtheappendixtoBookIII。]
Towhateversidemathematiciansturn，thisdilemmastillmeetsthem。Iftheyjudgeofequality，oranyotherproportion，bytheaccurateandexactstandard，viz。theenumerationoftheminuteindivisibleparts，theybothemployastandard，whichisuselessinpractice，andactuallyestablishtheindivisibilityofextension，whichtheyendeavourtoexplode。Oriftheyemploy，asisusual，theinaccuratestandard，deriv’dfromacomparisonofobjects，upontheirgeneralappearance，correctedbymeasuringandjuxtaposition；theirfirstprinciples，tho’certainandinfallible，aretoocoarsetoaffordanysuchsubtileinferencesastheycommonlydrawfromthem。Thefirstprinciplesarefoundedontheimaginationandsenses:Theconclusion，therefore，cannevergobeyond，muchlesscontradictthesefaculties。
Thismayopenoureyesalittle，andletussee，thatnogeometricaldemonstrationfortheinfinitedivisibilityofextensioncanhavesomuchforceaswhatwenaturallyattributetoeveryargument，whichissupportedbysuchmagnificentpretensions。Atthesametimewemaylearnthereason，whygeometryfallsofevidenceinthissinglepoint，，whileallitsotherreasoningscommandourfullestassentandapprobation。Andindeeditseemsmorerequisitetogivethereasonofthisexception，thantoshew，thatwereallymustmakesuchanexception，andregardallthemathematicalargumentsforinfinitedivisibilityasutterlysophistical。For’tisevident，thatasnoideaofquantityisinfinitelydivisible，therecannotbeimagin’damoreglaringabsurdity，thantoendeavourtoprove，thatquantityitselfadmitsofsuchadivision；andtoprovethisbymeansofideas，whicharedirectlyoppositeinthatparticular。Andasthisabsurdityisveryglaringinitself，sothereisnoargumentfoundedonit’。whichisnotattendedwithanewabsurdity，andinvolvesnotanevidentcontradiction。
Imightgiveasinstancesthoseargumentsforinfinitedivisibility，whicharederiv’dfromthepointofcontact。Iknowthereisnomathematician，whowillnotrefusetobejudg’dbythediagramshedescribesuponpaper，thesebeingloosedraughts，ashewilltellus，andservingonlytoconveywithgreaterfacilitycertainideas，whicharethetruefoundationofallourreasoning。ThisIamsatisfy’dwith，andamwillingtorestthecontroversymerelyupontheseideas。Idesirethereforeourmathematiciantoform，asaccuratelyaspossible，theideasofacircleandarightline；andIthenask，ifupontheconceptionoftheircontacthecanconceivethemastouchinginamathematicalpoint，orifhemustnecessarilyimaginethemtoconcurforsomespace。Whicheversidehechuses，herunshimselfintoequaldifficulties。Ifheaffirms，thatintracingthesefiguresinhisimagination，hecanimaginethemtotouchonlyinapoint，heallowsthepossibilityofthatidea，andconsequentlyofthething。Ifhesays，thatinhisconceptionofthecontactofthoselineshemustmakethemconcur，hetherebyacknowledgesthefallacyofgeometricaldemonstrations，whencarry’dbeyondacertaindegreeofminuteness；since’tiscertainhehassuchdemonstrationsagainsttheconcurrenceofacircleandarightline；thatis，inotherwords，becanproveanidea，viz。thatofconcurrence，tobeincompatiblewithtwootherideas，thoseofacircleandrightline；tho’atthesametimeheacknowledgestheseideastobeinseparable。
Thesamesubjectcontinued。
Ifthesecondpartofmysystembetrue，thattheideaofspaceorextensionisnothingbuttheideaofvisibleortangiblepointsdistributedinacertainorder；itfollows，thatwecanformnoideaofavacuum，orspace，wherethereisnothingvisibleortangible。’Thisgivesrisetothreeobjections，whichIshallexaminetogether，becausetheanswerIshallgivetooneisaconsequenceofthatwhichIshallmakeuseoffortheothers。
First，Itmaybesaid，thatmenhavedisputedformanyagesconcerningavacuumandaplenum，withoutbeingabletobringtheaffairtoafinaldecision；andphilosophers，evenatthisday，thinkthemselvesatlibertytotakepartoneitherside，astheirfancyleadsthem。Butwhateverfoundationtheremaybeforacontroversyconcerningthethingsthemselves，itmaybepretended，thattheverydisputeisdecisiveconcerningtheidea，andthat’tisimpossiblemencou’dsolongreasonaboutavacuum，andeitherrefuteordefendit，withouthavinganotionofwhattheyrefutedordefended。
Secondly，Ifthisargumentshou’dbecontested，therealityoratleastthepossibilityoftheideaofavacuummaybeprov’dbythefollowingreasoning。Everyideaispossible，whichisanecessaryandinfallibleconsequenceofsuchasarepossible。Nowtho’weallowtheworldtobeatpresentaplenum，wemayeasilyconceiveittobedepriv’dofmotion；andthisideawillcertainlybeallow’dpossible。Itmustalsobeallow’dpossible，toconceivetheannihilationofanypartofmatterbytheomnipotenceofthedeity，whiletheotherpartsremainatrest。Foraseveryidea，thatisdistinguishable，isseparablebytheimagination；andaseveryidea，thatisseparablebytheimagination，maybeconceiv’dtobeseparatelyexistent；’tisevident，thattheexistenceofoneparticleofmatter，nomoreimpliestheexistenceofanother，thanasquarefigureinonebodyimpliesasquarefigureineveryone。Thisbeinggranted，Inowdemandwhatresultsfromtheconcurrenceofthesetwopossibleideasofrestandannihilation，andwhatmustweconceivetofollowupontheannihilationofalltheairandsubtilematterinthechamber，supposingthewallstoremainthesame，withoutanymotionoralteration？Therearesomemetaphysicians，whoanswer，thatsincematterandextensionarethesame，theannihilationofonenecessarilyimpliesthatoftheother；andtherebeingnownodistancebetwixtthewallsofthechamber，theytoucheachother；inthesamemannerasmyhandtouchesthepaper，whichisimmediately’beforeme。Buttho’thisanswerbeverycommon，Idefythesemetaphysicianstoconceivethematteraccordingtotheirhypothesis，orimaginethefloorandroof，withalltheoppositesidesofthechamber，totoucheachother，whiletheycontinueinrest，andpreservethesameposition。Forhowcanthetwowalls，thatrunfromsouthtonorth，toucheachother，whiletheytouchtheoppositeendsoftwowalls，thatrunfromeasttowest？Andhowcanthefloorand。roofevermeet，whiletheyareseparatedbythefourwalls，thatlieinacontraryposition？Ifyouchangetheirposition，yousupposeamotion。Ifyouconceiveanythingbetwixtthem，yousupposeanewcreation。Butkeepingstrictlytothetwoideasofrestandannihilation，’tisevident，thattheidea，whichresultsfromthem，isnotthatofacontactofparts，butsomethingelse；whichisconcludedtobetheideaofavacuum。
Thethirdobjectioncarriesthematterstillfarther，andnotonlyasserts，thattheideaofavacuumisrealandpossible，butalsonecessaryandunavoidable。Thisassertionisfoundedonthemotionweobserveinbodies，which，’tismaintain’d，wou’dbeimpossibleandinconceivablewithoutavacuum，intowhichonebodymustmoveinordertomakewayforanother……Ishallnotenlargeuponthisobjection，becauseitprincipallybelongstonaturalphilosophy，whichlieswithoutourpresentsphere。
Inordertoanswertheseobjections，wemusttakethematterprettydeep，andconsiderthenatureandoriginofseveralideas，，lestwedisputewithoutunderstandingperfectlythesubjectofthecontroversy。’Tisevidenttheideaofdarknessisnopositiveidea，butmerelythenegationof。light，ormoreproperlyspeaking，ofcolour’dandvisibleobjects。Aman，whoenjoyshissight，receivesnootherperceptionfromturninghiseyesoneveryside，whenentirelydepriv’doflight，thanwhatiscommontohimwithonebornblind；and’tiscertainsuch-a-onehasnoideaeitheroflightordarkness。Theconsequenceofthisis，that’tisnotfromthemereremovalofvisibleobjectswereceivetheimpressionofextensionwithoutmatter；andthattheideaofutterdarknesscanneverbethesamewiththatofvacuum。
SupposeagainamantobeSupportedintheair，andtobesoftlyconvey’dalongbysomeinvisiblepower；’tisevident’heissensibleofnothing，andneverreceivestheideaofextension，norindeedanyidea，fromthisinvariablemotion。Evensupposinghemoveshislimbstoandfro，thiscannotconveytohimthatidea。Hefeelsinthatcaseacertainsensationorimpression，thepartsofwhicharesuccessivetoeachother，andmaygivehimtheideaoftime:Butcertainlyarenotdispos’dinsuchamanner，asisnecessarytoconveytheideaofsaceortheideaofspaceorextension。
Sincethenitappears，thatdarknessandmotion，withtheutterremovalofeverythingvisibleandtangible，cannevergiveustheideaofextensionwithoutmatter，orofavacuum；thenextquestionis，whethertheycanconveythisidea，whenmix’dwithsomethingvisibleandtangible？
’Tiscommonlyallow’dbyphilosophers，thatallbodies，whichdiscoverthemselvestotheeye，appearasifpaintedonaplainsurface，andthattheirdifferentdegreesofremotenessfromourselvesarediscoveredmorebyreasonthanbythesenses。WhenIholdupmyhandbeforeme，andspreadmyfingers，theyareseparatedasperfectlybythebluecolourofthefirmament，astheycou’dbebyanyvisibleobject，whichIcou’dplacebetwixtthem。Inorder，therefore，toknowwhetherthesightcanconveytheimpressionandideaofavacuum，wemustsuppose，thatamidstanentiredarkness，thereareluminousbodiespresentedtous，whoselightdiscoversonlythesebodiesthemselves，withoutgivingusanyimpressionofthesurroundingobjects。
Wemustformaparallelsuppositionconcerningtheobjectsofourfeeling。’Tienotpropertosupposeaperfectremovalofalltangibleobjects:wemustallowsomethingtobeperceiv’dbythefeeling；andafteranintervalandmotionofthehandorotherorganofsensation，anotherobjectofthetouchtobemetwith；anduponleavingthat，another；andsoon，asoftenasweplease。Thequestionis，whethertheseintervalsdonotaffordustheideaofextensionwithoutbody？
Tobeginwiththefirstcase；’tisevident，thatwhenonlytwoluminousbodiesappeartotheeye，wecanperceive，whethertheybeconjoin’dorseparate:whethertheybeseparatedbyagreatorsmalldistance；andifthisdistancevaries，wecanperceiveitsincreaseordiminution，withthemotionofthebodies。Butasthedistanceisnotinthiscaseanythingcolour’dorvisible，itmaybethoughtthatthereishereavacuumorpureextension，notonlyintelligibletothemind，butobvioustotheverysenses。
Thisisournaturalandmostfamiliarwayofthinking；butwhichweshalllearntocorrectbyalittlereflection。Wemayobserve，thatwhentwobodiespresentthemselves，wheretherewasformerlyanentiredarkness，theonlychange，thatisdiscoverable，isintheappearanceofthesetwoobjects，andthatalltherestcontinuestobeasbefore，aperfectnegationoflight，andofeverycolour’dorvisibleobject。Thisisnotonlytrueofwhatmaybesaidtoberemotefromthesebodies，butalsooftheverydistance；whichisinterposedbetwixtthem；thatbeingnothingbutdarkness，orthenegationoflight；withoutparts，withoutcomposition，invariableandindivisible。Nowsincethisdistancecausesnoperceptiondifferentfromwhatablindmanreceivesfromhiseyes，orwhatisconvey’dtousinthedarkestnight，itmustpartakeofthesameproperties:Andasblindnessanddarknessaffordusnoideasofextension，’tisimpossiblethatthedarkandundistinguishabledistancebetwixttwobodiescaneverproducethatidea。
Thesoledifferencebetwixtanabsolutedarknessandtheappearanceoftwoormorevisibleluminousobjectsconsists，asIsaid，intheobjectsthemselves，andinthemannertheyaffectoursenses。Theangles，whichtheraysoflightflowingfromthem，formwitheachother；themotionthatisrequir’dintheeye，initspassagefromonetotheother；andthedifferentpartsoftheorgans，whichareaffectedbythem；theseproducetheonlyperceptions，fromwhichwecanjudgeofthedistance。Butastheseperceptionsareeachofthemsimpleandindivisible，theycannevergiveustheideaofextension。
Wemayillustratethisbyconsideringthesenseoffeeling，andtheimaginarydistanceorintervalinterpos’dbetwixttangibleorsolidobjects。Isupposetwocases，viz。thatofamansupportedintheair，andmovinghislimbstoandfro，withoutmeetinganythingtangible；andthatofaman，whofeelingsomethingtangible，leavesit，andafteramotion，ofwhichheissensible，perceivesanothertangibleobject；andIthenask，whereinconsiststhedifferencebetwixtthesetwocases？Noonewillmakeanyscrupletoaffirm，thatitconsistsmeerlyintheperceivingthoseobjects，andthatthesensation，whicharisesfromthemotion，isinbothcasesthesame:Andasthatsensationisnotcapableofconveyingtousanideaofextension，whenunaccompany’dwithsomeotherperception，itcannomoregiveusthatidea，whenmix’dwiththeimpressionsoftangibleobjects；sincethatmixtureproducesnoalterationuponit。
Buttho’motionanddarkness，eitheralone，orattendedwithtangibleandvisibleobjects，conveynoideaofavacuumorextensionwithoutmatter，yettheyarethecauseswhywefalslyimaginewecanformsuchanidea。Forthereisacloserelation’betwixtthatmotionanddarkness，andarealextension，orcompositionofvisibleandtangibleobjects。
First，Wemayobserve，thattwovisibleobjectsappearinginthemidstofutterdarkness，affectthesensesinthesamemanner，andformthesameanglebytherays，whichflowfromthem，andmeetintheeye，asifthedistancebetwixtthemwerefindwithvisibleobjects，thatgiveusatrueideaofextension。Thesensationofmotionislikewisethesame，whenthereisnothingtangibleinterpos’dbetwixttwobodies，aswhenwefeelacompoundedbody，whosedifferentpartsareplac’dbeyondeachother。
Secondly，Wefindbyexperience，thattwobodies，whicharesoplac’dastoaffectthesensesinthesamemannerwithtwoothers，thathaveacertainextentofvisibleobjectsinterpos’dbetwixtthem，arecapableofreceivingthesameextent，withoutanysensibleimpulseorpenetration，andwithoutanychangeonthatangle，underwhichtheyappeartothesenses。Inlikemanner，wherethereisoneobject，whichwecannotfeelafteranotherwithoutaninterval，andtheperceivingofthatsensationwecallmotioninourhandororganofsensation；experienceshewsus，that’tispossiblethesameobjectmaybefeltwiththesamesensationofmotion，alongwiththeinterpos’dimpressionofsolidandtangibleobjects，attendingthesensation。Thatis，inotherwords，aninvisibleandintangibledistancemaybeconvertedintoavisibleandtangibleone，withoutanychangeonthedistantobjects。
Thirdly，Wemayobserve，asanotherrelationbetwixtthesetwokindsofdistance，thattheyhavenearlythesameeffectsoneverynaturalphaenomenon。Forasallqualities，suchasheat，cold，light，attraction，&；c。diminishinproportiontothedistance；thereisbutlittledifferenceobserv’d，whetherthisdistancebemarledoutbycompoundedandsensibleobjects，orbeknownonlybythemanner，inwhichthedistantobjectsaffectthesenses。
Herethenarethreerelationsbetwixtthatdistance，whichconveystheideaofextension，andthatother，whichisnotfill’dwithanycolour’dorsolidobject。Thedistantobjectsaffectthesensesinthesamemanner，whetherseparatedbytheonedistanceortheother；thesecondspeciesofdistanceisfoundcapableofreceivingthefirst；andtheybothequallydiminishtheforceofeveryquality。
Theserelationsbetwixtthetwokindsofdistancewillaffordusaneasyreason，whytheonehassooftenbeentakenfortheother，andwhyweimaginewehaveanideaofextensionwithouttheideaofanyobjecteitherofthesightorfeeling。Forwemayestablishitasageneralmaximinthisscienceofhumannature，thatwhereverthereisacloserelationbetwixttwoideas，themindisveryapttomistakethem，andinallitsdiscoursesandreasoningstousetheonefortheother。Thisphaenomenonoccursonsomanyoccasions，andisofsuchconsequence，thatIcannotforbearstoppingamomenttoexamineitscauses。Ishallonlypremise，thatwemustdistinguishexactlybetwixtthephaenomenonitself，andthecauses，whichIshallassignforit；andmustnotimaginefromanyuncertaintyinthelatter，thattheformerisalsouncertain。Thephaenomenonmaybereal，tho’myexplicationbechimerical。Thefalshoodoftheoneisnoconsequenceofthatoftheother；tho’atthesametimewemayobserve，that’tisverynaturalforustodrawsuchaconsequence；whichisanevidentinstanceofthatveryprinciple，whichIendeavourtoexplain。
WhenIreceiv’dtherelationsofresemblance，contiguityandcausation，asprinciplesofunionamongideas，withoutexaminingintotheircauses，’twasmoreinprosecutionofmyfirstmaxim，thatwemustintheendrestcontentedwithexperience，thanforwantofsomethingspeciousandplausible，whichImighthavedisplay’donthatsubject。’Twou’dhavebeeneasytohavemadeanimaginarydissectionofthebrain，andhaveshewn，whyuponourconceptionofanyidea，theanimalspiritsrunintoallthecontiguoustraces，androuzeuptheotherideas，thatarerelatedtoit。Buttho’Ihaveneglectedanyadvantage，whichImighthavedrawnfromthistopicinexplainingtherelationsofideas，IamafraidImustherehaverecoursetoit，inordertoaccountforthemistakesthatarisefromtheserelations。Ishallthereforeobserve，thatasthemindisendow’dwithapowerofexcitinganyideaitpleases；wheneveritdispatchesthespiritsintothatregionofthebrain，inwhichtheideaisplac’d；thesespiritsalwaysexcitetheidea，whentheyrunpreciselyintothepropertraces，andrummagethatcell，whichbelongstotheidea。Butastheirmotionisseldomdirect，andnaturallyturnsalittletotheonesideortheother；forthisreasontheanimalspirits，fallingintothecontiguoustraces，presentotherrelatedideasinlieuofthat，whichtheminddesir’datfirsttosurvey。Thischangewearenotalwayssensibleof；butcontinuingstillthesametrainofthought，makeuseoftherelatedidea，whichispresentedtous，andemployitinourreasoning，asifitwerethesamewithwhatwedemanded。Thisisthecauseofmanymistakesandsophismsinphilosophy；aswillnaturallybeimagin’d，andasitwou’dbeeasytoshow，iftherewasoccasion。
Ofthethreerelationsabove-mention’dthatofresemblanceisthemostfertilesourceoferror；andindeedtherearefewmistakesinreasoning，whichdonotborrowlargelyfromthatorigin。Resemblingideasarenotonlyrelatedtogether，buttheactionsofthemind，whichweemployinconsideringthem，aresolittledifferent，thatwearenotabletodistinguishthem。Thislastcircumstanceisofgreatconsequence，andwemayingeneralobserve，thatwherevertheactionsofthemindinforminganytwoideasarethesameorresembling，weareveryapttoconfoundtheseideas，andtaketheonefortheother。Ofthisweshallseemanyinstancesintheprogressofthistreatise。Buttho’resemblancebetherelation，whichmostreadilyproducesamistakeinideas，yettheothersofcausationandcontiguitymayalsoconcurinthesameinfluence。Wemightproducethefiguresofpoetsandorators，assufficientproofsofthis，wereitasusual，asitisreasonable，inmetaphysicalsubjectstodrawourargumentsfromthatquarter。Butlestmetaphysiciansshou’desteemthisbelowtheirdignity，Ishallborrowaprooffromanobservation，whichmaybemadeonmostoftheirowndiscourses，viz。that’tisusualformentousewordsforideas，andtotalkinsteadofthinkingintheirreasonings。Weusewordsforideas，becausetheyarecommonlysocloselyconnectedthatthemindeasilymistakesthem。Andthislikewiseisthereason，whywesubstitutetheideaofadistance，whichisnotconsideredeitherasvisibleortangible，intheroomofextension，whichisnothingbutacompositionofvisibleortangiblepointsdispos’dinacertainorder。Incausingthismistakethereconcurboththerelationsofcausationandresemblance。Asthefirstspeciesofdistanceisfoundtobeconvertibleintothesecond，’tisinthisrespectakindofcause；andthesimilarityoftheirmannerofaffectingthesenses，anddiminishingeveryquality，formstherelationofresemblance。
Afterthischainofreasoningandexplicationofmyprinciples，Iamnowprepar’dtoansweralltheobjectionsthathavebeenoffer’d，whetherderiv’dfrommetaphysicsormechanics。Thefrequentdisputesconcerningavacuum，orextensionwithoutmatterprovenottherealityoftheidea，uponwhichthedisputeturns；therebeingnothingmorecommon，thantoseemendeceivethemselvesinthisparticular；especiallywhenbymeansofanycloserelation，thereisanotherideapresented，whichmaybetheoccasionoftheirmistake。
Wemaymakealmostthesameanswertothesecondobjection，deriv’dfromtheconjunctionoftheideasofrestandannihilation。Wheneverythingisannihilatedinthechamber，andthewallscontinueimmoveable，thechambermustbeconceiv’dmuchinthesamemannerasatpresent，whentheairthatfillsit，isnotanobjectofthesenses。Thisannihilationleavestotheeye，thatfictitiousdistance，whichisdiscoveredbythedifferentpartsoftheorgan，thatareaffected，andbythedegreesoflightandshade；-andtothefeeling，thatwhichconsistsinasensationofmotioninthehand，orothermemberofthebody。Invainshou’dwe。searchanyfarther。Onwhicheversideweturnthissubject，weshallfindthatthesearetheonlyimpressionssuchanobjectcanproduceafterthesuppos’dannihilation；andithasalreadybeenremark’d，thatimpressionscangiverisetonoideas，buttosuchasresemblethem。
Sinceabodyinterposedbetwixttwoothersmaybesuppos’dtobeannihilated，withoutproducinganychangeuponsuchaslieoneachhandofit，’tiseasilyconceiv’d，howitmaybecreatedanew，andyetproduceaslittlealteration。Nowthemotionofabodyhasmuchthesameeffectasitscreation。Thedistantbodiesarenomoreaffectedintheonecase，thanintheother。Thissufficestosatisfytheimagination，andprovesthereisnorepugnanceinsuchamotion。Afterwardsexperiencecomesinplaytopersuadeusthattwobodies，situatedinthemannerabove-describ’d，havereallysuchacapacityofreceivingbodybetwixtthem，andthatthereisnoobstacletotheconversionoftheinvisibleandintangibledistanceintoonethatisvisibleandtangible。Howevernaturalthatconversionmayseem，wecannotbesureitispracticable，beforewehavehadexperienceofit。
ThusIseemtohaveanswer’dthethreeobjectionsabove-mention’d；tho’atthesametimeIamsensible，thatfewwillbesatisfy’dwiththeseanswers，butwillimmediatelyproposenewobjectionsanddifficulties。’Twillprobablybesaid，thatmyreasoningmakesnothingtothematterinhandsandthatIexplainonlythemannerinwhichobjectsaffectthesenses，withoutendeavouringtoaccountfortheirrealnatureandoperations。Tho’therebenothingvisibleortangibleinterposedbetwixttwobodies，yetwefindbyexperience，thatthebodiesmaybeplac’dinthesamemanner，withregardtotheeye，andrequirethesamemotionofthehandinpassingfromonetotheother，asifdividedbysomethingvisibleandtangible。Thisinvisibleandintangibledistanceisalsofoundbyexperiencetocontainacapacityofreceivingbody，orofbecomingvisibleandtangible。Hereisthewholeofmysystem；andinnopartofithaveIendeavour’dtoexplainthecause，whichseparatesbodiesafterthismanner，andgivesthemacapacityofreceivingothersbetwixtthem，withoutanyimpulseorpenetration。
Ianswerthisobjection，bypleadingguilty，andbyconfessingthatmyintentionneverwastopenetrateintothenatureofbodies，orexplainthesecretcausesoftheiroperations。Forbesidesthatthisbelongsnottomypresentpurpose，Iamafraid，thatsuchanenterpriseisbeyondthereachofhumanunderstanding，andthatwecanneverpretendtoknowbodyotherwisethanbythoseexternalproperties，whichdiscoverthemselvestothesenses。Astothosewhoattemptanythingfarther，Icannotapproveoftheirambition，tillIsee，insomeoneinstanceatleast，thattheyhavemetwithsuccess。ButatpresentIcontentmyselfwithknowingperfectlythemannerinwhichobjectsaffectmysenses，andtheirconnectionswitheachother，asfarasexperienceinformsmeofthem。Thissufficesfortheconductoflife；andthisalsosufficesformyphilosophy，whichpretendsonlytoexplainthenatureandcausesofourperceptions，orimpressionsandideas。(11)
Ishallconcludethissubjectofextensionwithaparadox，whichwilleasilybeexplain’dfromtheforegoingreasoning。Thisparadoxis，thatifyouarepleas’dtogivetothein-visibleandintangibledistance，orinotherwords，tothecapacityofbecomingavisibleandtangibledistance，thenameofavacuum，extensionandmatterarethesame，andyetthereisavacuum。Ifyouwillnotgiveitthatname，motionispossibleinaplenum，withoutanyimpulseininfinitum，withoutreturninginacircle，andwithoutpenetration。Buthoweverwemayexpressourselves，wemustalwaysconfess，thatwehavenoideaofanyrealextensionwithoutfillingitwithsensibleobjects，andconceivingitspartsasvisibleortangible。
Astothedoctrine，thattimeisnothingbutthemanner，inwhichsomerealobjectsexist；wemayobserve，that’tisliabletothesameobjectionsasthesimilardoctrinewithregardtoextension。Ifitbeasufficientproof，thatwehavetheideaofavacuum，becausewedisputeandreasonconcerningit；wemustforthesamereasonhavetheideaoftimewithoutanychangeableexistence；sincethereisnosubjectofdisputemorefrequentandcommon。’Butthatwereallyhavenosuchidea，iscertain。Forwhenceshou’ditbederiv’d？Doesitarisefromanimpressionofsensationorofreflection？Pointitoutdistinctlytous，thatwemayknowitsnatureandqualities。Butifyoucannotpointoutanysuchimpression，youmaybecertainyouaremistaken，whenyouimagineyouhaveanysuchidea。
Buttho’itbeimpossibletoshewtheimpression，fromwhichtheideaoftimewithoutachangeableexistenceisderiv’d；yetwecaneasilypointoutthoseappearances，whichmakeusfancywehavethatidea。Forwemayobserve，thatthereisacontinualsuccessionofperceptionsinourmind；sothattheideaoftimebeingforeverpresentwithus；whenweconsiderastedfastobjectatfive-a-clock，andregardthesameatsix；weareapttoapplytoitthatideainthesamemannerasifeverymomentweredistinguish’dbyadifferentposition，oranalterationoftheobject。The:firstandsecondappearancesoftheobject，beingcompar’dwiththesuccessionofourperceptions，seemequallyremov’dasiftheobjecthadreallychang’d。Towhichwemayadd，whatexperienceshewsus，thattheobjectwassusceptibleofsuchanumberofchangesbetwixttheseappearances；asalsothattheunchangeableorratherfictitiousdurationhasthesameeffectuponeveryquality，byencreasingordiminishingit，asthatsuccession，whichisobvioustothesenses。Fromthesethreerelationsweareapttoconfoundourideas，andimaginewecanformtheideaofatimeandduration，withoutanychangeorsuccession。
OftheIdeaofExistence，andofExternalExistence。
Itmaynotbeamiss，beforeweleavethissubject，toexplaintheideasofexistenceandofexternalexistence；whichhavetheirdifficulties，aswellastheideasofspaceandtime。Bythismeansweshallbethebetterprepar’dfortheexaminationofknowledgeandprobability，whenweunderstandperfectlyallthoseparticularideas，whichmayenterintoourreasoning。
Thereisnoimpressionnorideaofanykind，ofwhichwehaveanyconsciousnessormemory，thatisnotconceiv’dasexistent；and’tisevident，thatfromthisconsciousnessthemostperfectideaandassuranceofbeingisderiv’d。Fromhencewemayformadilemma，themostclearandconclusivethatcanbeimagin’d，viz。thatsinceweneverrememberanyideaorimpressionwithoutattributingexistencetoit，theideaofexistencemusteitherbederiv’dfromadistinctimpression，conjoin’dwitheveryperceptionorobjectofourthought，ormustbetheverysamewiththeideaoftheperceptionorobject。
Asthisdilemmaisanevidentconsequenceoftheprinciple，thateveryideaarisesfromasimilarimpression，soourdecisionbetwixtthepropositionsofthedilemmaisnomoredoubtful。gofarfromtherebeinganydistinctimpression，attendingeveryimpressionandeveryidea，thatIdonotthinkthereareanytwodistinctimpressions，whichareinseparablyconjoin’d。Tho’certainsensationsmayatonetimebeunited，wequicklyfindtheyadmitofaseparation，andmaybepresentedapart。Andthus，tho’everyimpressionandideawerememberbeconsideredasexistent，theideaofexistenceisnotderiv’dfromanyparticularimpression。
Theideaofexistence，then，istheverysamewiththeideaofwhatweconceivetobeexistent。Toreflectonanythingsimply，andtoreflectonitasexistent，arenothingdifferentfromeachother。Thatidea，whenconjoin’dwiththeideaofanyobject，makesnoadditiontoit。Whateverweconceive，weconceivetobeexistent。Anyideawepleasetoformistheideaofabeing；andtheideaofabeingisanyideawepleasetoform。’
Whoeveropposesthis，mustnecessarilypointoutthatdistinctimpression，fromwhichtheideaofentityisderiv’d，andmustprove，thatthisimpressionisinseparablefromeveryperceptionwebelievetobeexistent。Thiswemaywithouthesitationconcludetobeimpossible。
Ourforegoing(12)reasoningconcerningthedistinctionofideaswithoutanyrealdifferencewillnothereserveusinanystead。Thatkindofdistinctionisfoundedonthedifferentresemblances，whichthesamesimpleideamayhavetoseveraldifferentideas。Butnoobjectcanbepresentedresemblingsomeobjectwithrespecttoitsexistence，anddifferentfromothersinthesameparticular；sinceeveryobject，thatispresented，mustnecessarilybeexistent。
Alikereasoningwillaccountfortheideaofexternalexistence。Wemayobserve，that’tisuniversallyallow’dbyphilosophers，andisbesidesprettyobviousofitself，thatnothingiseverreallypresentwiththemindbutitsperceptionsorimpressionsandideas，andthatexternalobjectsbecomeknowntousonlybythoseperceptionstheyoccasion。Tohate，tolove，tothink，tofeel，tosee；allthisisnothingbuttoperceive。
Nowsincenothingiseverpresenttothemindbutperceptions，andsinceallideasarederiv’dfromsomethingantecedentlypresenttothemind；itfollows，that’tisimpossibleforussomuchastoconceiveorformanideaofanythingspecificallydifferent。fromideasandimpressions。Letusfixourattentionoutofourselvesasmuchaspossible:Letuschaseourimaginationtotheheavens，ortotheutmostlimitsoftheuniverse；weneverreallyadvanceastepbeyondourselves，norcanconceiveanykindofexistence，butthoseperceptions，whichhaveappear’dinthatnarrowcompass。Thisistheuniverseoftheimagination，norhaveweanyideabutwhatisthereproduc’d。
Thefarthestwecangotowardsaconceptionofexternalobjects，whensuppos’dspecificallydifferentfromourperceptions，istoformarelativeideaofthem，withoutpretendingtocomprehendtherelatedobjects。Generallyspeakingwedonotsupposethemspecificallydifferent；butonlyattributetothemdifferentrelations，connectionsanddurations。Butofthismorefullyhereafter。(13)
OFKNOWLEDGEANDPROBABILITY。
OfKnowledge。
Thereareseven(14)differentkindsofphilosophicalrelation，viz。resemblance，identity，relationsoftimeandplace，proportioninquantityornumber，degreesinanyquality，contrarietyandcausation。Theserelationsmaybedividedintotwoclasses；intosuchasdependentirelyontheideas，whichwecomparetogether，andsuchasmaybechang’dwithoutanychangeintheideas。’Tisfromtheideaofatriangle，thatwediscovertherelationofequality，whichitsthreeanglesbeartotworightones；andthisrelationisinvariable，aslongasouridearemainsthesame。Onthecontrary，therelationsofcontiguityanddistancebetwixttwoobjectsmaybechang’dmerelybyanalterationoftheirplace，withoutanychangeontheobjectsthemselvesorontheirideas；andtheplacedependsonahundreddifferentaccidents，whichcannotbeforeseenbythemind。’Tisthesamecasewithidentityandcausation。Twoobjects，tho’perfectlyresemblingeachother，andevenappearinginthesameplaceatdifferenttimes，maybenumericallydifferent:Andasthepower，bywhichoneobjectproducesanother，isneverdiscoverablemerelyfromtheiridea，’tisevidentcauseandeffectarerelations，ofwhichwereceiveinformationfromexperience，andnotfromanyabstractreasoningorreflection。Thereisnosinglephaenomenon，eventhemostsimple，whichcanbeaccountedforfromthequalitiesoftheobjects，astheyappeartous；orwhichwecou’dforeseewithoutthehelpofourmemoryandexperience。
Itappears，therefore，thatofthesesevenphilosophicalrelations，thereremainonlyfour，whichdependingsolelyuponideas，canbetheobjectsofknowledgesaidcertainty。Thesefourareresemblance，contrariety，degreesinquality，andproportionsinquantityornumber。Threeoftheserelationsarediscoverableatfirstsight，andfallmoreproperlyundertheprovinceofintuitionthandemonstration。Whenanyobjectsresembleeachother，theresemblancewillatfirststriketheeve，orratherthemind；andseldomrequiresasecondexamination。Thecaseisthesamewithcontrariety，andwiththedegreesofanyquality。Noonecanoncedoubtbutexistenceandnon-existencedestroyeachother，andareperfectlyincompatibleandcontrary。Andtho’itbeimpossibletojudgeexactlyofthedegreesofanyquality，suchascolour，taste，heat，cold，whenthedifferencebetwixtthemisverysmall:yet’tiseasytodecide，thatanyofthemissuperiororinferiortoanother，whentheirdifferenceisconsiderable。Andthisdecisionwealwayspronounceatfirstsight，withoutanyenquiryorreasoning。
Wemightproceed，afterthesamemanner，infixingtheproportionsofquantityornumber，andmightatoneviewobserveasuperiorityorinferioritybetwixtanynumbers，orfigures；especiallywherethedifferenceisverygreatandremarkable。Astoequalityoranyexactproportion，wecanonlyguessatitfromasingleconsideration；exceptinveryshortnumbers，orverylimitedportionsofextension；whicharecomprehendedinaninstant，andwhereweperceiveanimpossibilityoffallingintoanyconsiderableerror。Inallothercaseswemustsettletheproportionswithsomeliberty，orproceedinamoreartificialmanner。
IhavealreadyIobserv’d’，thatgeometry，ortheart，bywhichwefixtheproportionsoffigures；tho’itmuchexcelsbothinuniversalityandexactness，theloosejudgmentsofthesensesandimagination；yetneverattainsaperfectprecisionandexactness。It’sfirstprinciplesarestilldrawnfromthegeneralappearanceoftheobjects；andthatappearancecanneveraffordusanysecurity，whenweexamine，theprodigiousminutenessofwhichnatureissusceptible。Ourideasseemtogiveaperfectassurance，thatnotworightlinescanhaveacommonsegment；butifweconsidertheseideas，weshallfind，thattheyalwayssupposeasensibleinclinationofthetwolines，andthatwheretheangletheyformisextremelysmall，wehavenostandardofaI@rightlinesopreciseastoassureusofthetruthofthisproposition。’Tisthesamecasewithmostoftheprimarydecisionsofthemathematics。
Thereremain，therefore，algebraandarithmeticastheonlysciences，inwhichwecancarryonachainofreasoningtoanydegreeofintricacy，andyetpreserveaperfectexactnessandcertainty。Wearepossestofaprecisestandard，bywhichwecanjudgeoftheequalityandproportionofnumbers；andaccordingastheycorrespondornottothatstandard，wedeterminetheirrelations，withoutanypossibilityoferror。’Whentwonumbersaresocombin’d，asthattheonehasalwaysanuniteansweringtoeveryuniteoftheother，wepronouncethemequal；and’tisforwantofsuchastandardofequalityinextension，thatgeometrycanscarcebeesteem’daperfectandinfalliblescience。
Buthereitmaynotbeamisstoobviateadifficulty，whichmayarisefrommyasserting，thattho’geometryfallsshortofthatperfectprecisionandcertainty，whicharepeculiartoarithmeticandalgebra，yetitexcelstheimperfectjudgmentsofoursensesandimagination。ThereasonwhyIimputeanydefecttogeometry，is，becauseitsoriginalandfundamentalprinciplesarederiv’dmerelyfromappearances；anditmayperhapsbeimagin’d，thatthisdefectmustalwaysattendit，andkeepitfromeverreachingagreaterexactnessinthecomparisonofobjectsorideas，thanwhatoureyeorimaginationaloneisabletoattain。Iownthatthisdefectsofarattendsit，astokeepitfromeveraspiringtoafullcertainty:Butsincethesefundamentalprinciplesdependontheeasiestandleastdeceitfulappearances，theybestowontheirconsequencesadegreeofexactness，ofwhichtheseconsequencesaresinglyincapable。’Tisimpossiblefortheeyetodeterminetheanglesofachiliagontobeequalto1996rightangles，ormakeanyconjecture，thatapproachesthisproportion；butwhenitdetermines，thatrightlinescannotconcur；thatwecannotdrawmorethanonerightlinebetweentwogivenpoints；it’smistakescanneverbeofanyconsequence。Andthisisthenatureanduseofgeometry，torunusuptosuchappearances，as，byreasonoftheirsimplicity，cannotleadusintoanyconsiderableerror。
Ishallheretakeoccasiontoproposeasecondobservationconcerningourdemonstrativereasonings，whichissuggestedbythesamesubjectofthemathematics。’Tisusualwithmathematicians，topretend，thatthoseideas，whicharetheirobjects，areofsorefin’dandspiritualanature，thattheyfallnotundertheconceptionofthefancy，butmustbecomprehendedbyapureandintellectualview，ofwhichthesuperiorfacultiesofthesoularealonecapable。Thesamenotionrunsthro’mostpartsofphilosophy，andisprincipallymadeuseoftoexplainoarabstractideas，andtoshewhowwecanformanideaofatriangle，forinstance，whichshallneitherbeanisocelesnorscalenum，norbeconfin’dtoanyparticularlengthandproportionofsides。’Tiseasytosee，whyphilosophersaresofondofthisnotionofsomespiritualandrefin’dperceptions；sincebythatmeanstheycovermanyoftheirabsurdities，andmayrefusetosubmittothedecisionsofclearideas，byappealingtosuchasareobscureanduncertain。Buttodestroythisartifice，weneedbutreflectonthatprinciplesooftinsistedon，thatallourideasarecopy’dfromourimpressions。Forfromthencewemayimmediatelyconclude，thatsinceallimpressionsareclearandprecise，theideas，whicharecopy’dfromthem，mustbeofthesamenature，andcannever，butfromourfault，containanythingsodarkandintricate。Anideaisbyitsverynatureweakerandfainterthananimpression；butbeingineveryotherrespectthesame，cannotimplyanyverygreatmystery。’Ifitsweaknessrenderitobscure，’tisourbusinesstoremedythatdefect，asmuchaspossible，bykeepingtheideasteadyandprecise；andtillwehavedoneso，’tisinvaintopretendtoreasoningandphilosophy。
