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1、NovaActaEruditorumpp.369-372;Aug..1733.L.Euler.1ConstructioAequationumquarundamdifferentialium…..E011:Translated&AnnotatedbyIanBruce.L.EULER.TheTransformationofcertainDifferentialEquations,whichdonotpermittheseparationofvariables.Therearetwokindsofconstructionsusedtodeterminethem
2、agnitudeofsomeunknownquantityfromgeometry;totheformercanbereferredallgeometricalconstructionssuchasthoseinvolvingplanes,solidshapesandcurves;thelattertrulyarerelatetothesetransformationsbymeansofwhicheitherthequadratureorrectificationofcurvescanbebroughtabout.Weusethesetransforma
3、tionsincommongeometry,inorderthattherootsofsomekindofalgebraicequationcanbeexpressed;andthisisbroughtaboutbytheintesectionofcurvesorstraightlinesthatwehaveagreedupontouse,astheequationsetoutrequires.Trulythetransformationsofthelatterkind,whichcanbecalledtranscendental,areusedinth
4、eresolutionofdifferentialequations,whichcannotbetransformedintoalgebraicequations.Moreoverequations,whitheralgebraicortranscendental,towhichtwovariablequantitiesbelong,requiretransformationsofthiskindinorderthat,withsomevaluetopleasesetforonevariable,theothercanbedetermined.Inord
5、erthatthiscanbebroughtaboutforalgebraicequations,itisestablishedasapostulatethatsomealgebraicfunctionZcanbeproducedforagivenmagnitudez.Moreoverfortheabovedifferentialortranscendentalequations,∫Zdzistobedefinedanew,inwhichZdenotessomefunctionorotherofz,eitheralgebraicortranscenden
6、tal,andassuchcanbeconsideredwiththegivenequation.Thereforeonaccountofthis,aproposedequationcanoftenthusbetransformed,inorderthatitcanbeequaltotheothervariable,ortosomefunctionofthisvariable,bywhichthewholetransformationofthatequationwillbemadeevident.Moreoveritiscustomerytocallan
7、equationofthetransformedvariableseparated;fromwhichitislikewisetobeunderstood,thatacarefulseparationmadeinthismannerisalwaysrequiredfortranscendentalequations.Howeverthisseparationisnotrequiredinthesettingoutofalgebraicequations.Indeed,forthesewiththevariablesmixedtogether,thewho
8、lebusinesscanbeequallyeasytocomplete.The
