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This,,,becomes,,,progressively,,,more,,,elaborate,,,as,,,we,,,go,,,to,,,higher,,,orders;,,,finally,,,,the,,,even,,,and,,,odd,,,properties,,,of,,,functions,,,are,,,exploited,,,to,,,find,,,new,,,functions,,,associated,,,with,,,two,,,abscissas,,,,leading,,,in,,,one,,,example,,,to,,,a,,,constant,,,product,,,of,,,the,,,applied,,,lines,,,,which,,,are,,,generalized,,,in,,,turn.,,,It,,,is,,,of,,,interest,,,to,,,see,,,how,,,Euler,,,handled,,,these,,,shapes,,,,such,,,as,,,the,,,different,,,kinds,,,of,,,ellipsoid,,,,paraboloid,,,,and,,,hyperboloid,,,in,,,three,,,dimensional,,,diagrams,,,,together,,,with,,,their,,,cross-sections,,,and,,,asymptotic,,,cones,,,,where,,,appropriate.,,,American,Mathematical,Monthly,,76,(7),,pp.,This,,chapter,,still,,has,,meaning,,for,,the,,math.,,After,,some,,initial,,spade,,work,,,Euler,,shows,,how,,to,,express,,such,,asymptotic,,curves,,in,,terms,,of,,variables,,related,,to,,straight,,line,,asymptotes;,,a,,systematic,,method,,is,,then,,developed,,for,,dealing,,with,,curves,,up,,the,,fourth,,order,,,showing,,their,,asymptotic,,nature,,,but,,which,,could,,be,,extended,,easily,,to,,higher,,orders.,,This,,,is,,,another,,,long,,,and,,,thoughtful,,,chapter,,,,in,,,which,,,Euler,,,investigates,,,types,,,of,,,curves,,,both,,,with,,,and,,,without,,,diameters;,,,the,,,coordinates,,,chosen,,,depend,,,on,,,the,,,particular,,,symmetry,,,of,,,the,,,curve,,,,considered,,,algebraic,,,and,,,closed,,,with,,,a,,,finite,,,number,,,of,,,equal,,,parts.,,,Euler,shows,how,both,orthogonal,and,skew,coordinate,systems,may,be,changed,,both,by,changing,the,origin,and,by,rotation,,for,the,same,curve.,

The,largest,root,can,be,found,from,the,ratio,of,succeeding,terms,,etc.,Dutka,J.,,The,early,history,of,the,factorial,function.,91-136,(1991).,Click,here,for,the,5th,Appendix,:,Surfaces,of,the,second,order.,Click,,here,,for,,the,,2nd,,Chapter,,:,,About,,changing,,coordinates.,,Thus,,Euler,,ends,,this,,work,,in,,mid-stream,,as,,it,,were,,,as,,in,,his,,other,,teaching,,texts,,,as,,there,,was,,no,,final,,end,,to,,his,,machinations,,ever.,,Click,,,here,,,for,,,the,,,1st,,,Chapter,,,:,,,Concerning,,,the,,,kinds,,,of,,,functions.,,,Wanner,,(1996),,Analysis,,by,,its,,History,,,chapter,,1,,,pp,,1,,to,,79,,,Undergraduate,,Texts,,in,,Mathematics,,#70,,,ISBN,,978-0-387-77036-9,,MR1410751,,.,,Praefatio.,To,,,this,,,theory,,,,another,,,more,,,sophisticated,,,approach,,,is,,,appended,,,finally,,,,giving,,,the,,,same,,,results.,,,

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