第486章

第486章forthehumanmindtheabsolutecontinuityofmotionisinconceivable.thelawsofmotionofanykindonlybecomecomprehensibletomanwhenheexaminesunitsofthismotion,arbitrarilyselected.butatthesametimeitisfromthisarbitrarydivisionofcontinuousmotionintodiscontinuousunitsthatagreatnumberofhumanerrorsproceeds.

weallknowtheso-calledsophismoftheancients,provingthatachilleswouldneverovertakethetortoise,thoughachilleswalkedtentimesasfastasthetortoise.assoonasachillespassesoverthespaceseparatinghimfromthetortoise,thetortoiseadvancesone-tenthofthatspace：achillespassesoverthattenth,butthetortoisehasadvancedahundredth,andsoontoinfinity.thisproblemseemedtotheancientsinsoluble.theirrationalityoftheconclusion（thatachilleswillneverovertakethetortoise）arisesfromthearbitraryassumptionofdisconnectedunitsofmotion,whenthemotionbothofachillesandthetortoisewascontinuous.

bytakingsmallerandsmallerunitsofmotionwemerelyapproachthesolutionoftheproblem,butweneverattainit.itisonlybyassuminganinfinitelysmallmagnitude,andaprogressionrisingfromituptoatenth,andtakingthesumofthatgeometricalprogression,thatwecanarriveatthesolutionoftheproblem.anewbranchofmathematics,dealingwithinfinitelysmallquantities,givesnowinothermorecomplexproblemsofdynamicssolutionsofproblemsthatseemedinsoluble.

thisnewbranchofmathematics,unknowntotheancients,byassuminginfinitelysmallquantities,thatis,suchassecurethechiefconditionofmotion（absolutecontinuity）,correctstheinevitableerrorwhichthehumanintellectcannotbutmake,whenitconsidersdisconnectedunitsofmotioninsteadofcontinuousmotion.

intheinvestigationofthelawsofhistoricalmotionpreciselythesamemistakearises.

theprogressofhumanity,arisingfromaninnumerablemultitudeofindividualwills,iscontinuousinitsmotion.

thediscoveryofthelawsofthismotionistheaimofhistory.butinordertoarriveatthelawsofthecontinuousmotionduetothesumofalltheseindividualwills,thehumanmindassumesarbitrary,disconnectedunits.thefirstproceedingofthehistorianistakinganarbitraryseriesofcontinuouseventstoexamineitapartfromothers,whileinrealitythereisnot,andcannotbe,abeginningtoanyevent,butoneeventflowswithoutanybreakincontinuityfromanother.thesecondproceedingistoexaminetheactionofasingleperson,asovereign,orageneral,asthoughitwereequivalenttothesumofmanyindividualwills,thoughthesumofindividualwillsneverfindsexpressionintheactionofasinglehistoricalpersonage.

historicalscienceasitadvancesiscontinuallytakingsmallerandsmallerunitsforanalysis,andinthiswaystrivestoapproximatethetruth.buthoweversmalltheunitsofwhichhistorytakescognisance,wefeelthattheassumptionofaunit,disconnectedfromanother,theassumptionofabeginningofanyphenomenon,andtheassumptionthattheindividualwillsofallmenfindexpressionintheactionsofasinglehistoricalpersonagearefalseinthemselves.

everyconclusionofhistorycan,withouttheslightesteffortonthepartofthecritic,bedissipatedlikedust,leavingnotrace,simplythroughcriticismselecting,astheobjectofitsanalysis,agreaterorsmallerdisconnectedunit,whichithasaperfectrighttodo,seeingthattheunitofhistoryisalwaysselectedarbitrarily.

onlybyassuminganinfinitelysmallunitforobservation—adifferentialofhistory—thatis,thehomogeneoustendenciesofmen,andarrivingattheintegralcalculus（thatis,takingthesumofthoseinfinitesimalquantities）,canwehopetoarriveatthelawsofhistory.