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1、FoundationsofDifferentialCalculusEulerSpringerPrefaceWhatdifferentialcalculus,and,ingeneral,analysisoftheinfinite,mightbecanhardlybeexplainedtothoseinnocentofanyknowledgeofit.Norcanwehereofferadefinitionatthebeginningofthisdissertationasissometimesdoneinotherdisciplines.Itisnotthatthereisnoc
2、leardefinitionofthiscalculus;rather,thefactisthatinordertounderstandthedefinitionthereareconceptsthatmustfirstbeunderstood.Besidesthoseideasincommonusage,therearealsoothersfromfiniteanalysisthataremuchlesscommonandareusuallyexplainedinthecourseofthedevelopmentofthedifferentialcalculus.Forthis
3、reason,itisnotpossibletounderstandadefinitionbeforeitsprinciplesaresufficientlyclearlyseen.Inthefirstplace,thiscalculusisconcernedwithvariablequantities.Althougheveryquantitycannaturallybeincreasedordecreasedwithoutlimit,still,sincecalculusisdirectedtoacertainpurpose,wethinkofsomequantitiesa
4、sbeingconstantlythesamemagnitude,whileotherschangethroughallthestagesofincreasinganddecreasing.Wenotethisdistinc-tionandcalltheformerconstantquantitiesandthelattervariables.Thischaracteristicdifferenceisnotrequiredbythenatureofthings,butratherbecauseofthespecialquestionaddressedbythecalcu
5、lus.Inorderthatthisdifferencebetweenconstantquantitiesandvariablesmightbeclearlyillustrated,letusconsiderashotfiredfromacannonwithachargeofgunpowder.Thisexampleseemstobeespeciallyappropriatetoclarifythismatter.Therearemanyquantitiesinvolvedhere:First,thereisthequantityofgunpowder;then,thea
6、ngleofelevationofthecannonabovethehorizon;third,thedistancetraveledbytheshot;and,fourth,thelengthviPrefaceoftimetheshotisintheair.Unlessthesamecannonisusedthroughouttheexperiment,wemustalsobringintoourcalculationsthelengthofthebarrelandtheweightoftheshot.Here,wewillnotconsidervariationsi
7、nthecannonortheshot,lestwebecomeentailedinverycomplicatedquestions.Hence,ifwealwayskeepthesamequantityofpowder,theelevationofthebarrelwillvarycontinuouslywiththedistancetraveledandtheshot’sdurationoftimeintheair.Inthiscase,theamountofpowder,ortheforceoftheexplosion,willbe
