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1、FoundationsofDifferentialCalculusEulerSpringerPrefaceWhatdifferentialcalculus,and,ingeneral,analysisoftheinfinite,mightbecanhardlybeexplainedtothoseinnocentofanyknowledgeofit.Norcanwehereofferadefinitionatthebeginningofthisdissertationasissometimesdoneinotherdisciplines.Itisnotthattherei
2、snocleardefinitionofthiscalculus;rather,thefactisthatinordertounderstandthedefinitionthereareconceptsthatmustfirstbeunderstood.Besidesthoseideasincommonusage,therearealsoothersfromfiniteanalysisthataremuchlesscommonandareusuallyexplainedinthecourseofthedevelopmentofthedifferentialcalculus
3、.Forthisreason,itisnotpossibletounderstandadefinitionbeforeitsprinciplesaresufficientlyclearlyseen.Inthefirstplace,thiscalculusisconcernedwithvariablequantities.Althougheveryquantitycannaturallybeincreasedordecreasedwithoutlimit,still,sincecalculusisdirectedtoacertainpurpose,wethinkofsom
4、equantitiesasbeingconstantlythesamemagnitude,whileotherschangethroughallthestagesofincreasinganddecreasing.Wenotethisdistinc-tionandcalltheformerconstantquantitiesandthelattervariables.Thischaracteristicdifferenceisnotrequiredbythenatureofthings,butratherbecauseofthespecialquestionadd
5、ressedbythecalculus.Inorderthatthisdifferencebetweenconstantquantitiesandvariablesmightbeclearlyillustrated,letusconsiderashotfiredfromacannonwithachargeofgunpowder.Thisexampleseemstobeespeciallyappropriatetoclarifythismatter.Therearemanyquantitiesinvolvedhere:First,thereisthequantityo
6、fgunpowder;then,theangleofelevationofthecannonabovethehorizon;third,thedistancetraveledbytheshot;and,fourth,thelengthviPrefaceoftimetheshotisintheair.Unlessthesamecannonisusedthroughouttheexperiment,wemustalsobringintoourcalculationsthelengthofthebarrelandtheweightoftheshot.Here,wewi
7、llnotconsidervariationsinthecannonortheshot,lestwebecomeentailedinverycomplicatedquestions.Hence,ifwealwayskeepthesamequantityofpowder,theelevationofthebarrelwillvarycontinuouslywiththedistancetraveledandtheshot’sdurationoftimeintheair.Inthiscase,theamountofpowder,ortheforceoftheexpl
8、osion,willbe
