Caverni, Raffaello, Storia del metodo sperimentale in Italia, 1891-1900

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            <p type="caption">
              <s>
                <pb xlink:href="020/01/2817.jpg" pagenum="442"/>
              medesimo tempo “ ut ergo omnes tangentes curvae AF ad omnes tangentes
                <lb/>
              arcus IMF, sic ipsa curva AF ad ipsum arcum IMF (pag. </s>
              <s>341). </s>
            </p>
            <p type="main">
              <s>A instituire i terzo principio, essendo FG tangente al circolo nel punto
                <lb/>
              F, FH tangente alla curva, e perciò resultante del moto, si conducano il rag­
                <lb/>
              gio FL, e la corda FI. </s>
              <s>I triangoli simili FGH, FLI danno FH ad FG, come
                <lb/>
              IF ad FL. </s>
              <s>Se ora intendansi fatte nell'arco IMF, e nella porzione di curva
                <lb/>
              AF, le medesime infinite divisioni, e, condotte le medesime infinite tangenti,
                <lb/>
              se ne prenda le somme; ne concluderemo, con l'Autore, per terzo principio,
                <lb/>
              “ chordas illas omnes simul sumptas, ad radium FL toties sumptum, sic se
                <lb/>
              habere, ut omnes tangentes curvae AF simul, ad omnes tangentes arcus IMF
                <lb/>
              simul, hoc est, per secundum notatum, ut curva ipsa AF, ad arcum ipsum
                <lb/>
              IMF ” (pag. </s>
              <s>341). </s>
            </p>
            <p type="main">
              <s>Premonstrati i quali principii, così facilmente si conduce il Roberval alla
                <lb/>
              desiderata conclusione. </s>
              <s>Dagli infiniti punti di divisione dell'arco IM, metà di
                <lb/>
              IMF, si conducano sul raggio LI gl'infiniti seni retti corrispondenti, ciascun
                <lb/>
              de'quali essendo la metà della corda, la metà pure sarà quella di questa loro
                <lb/>
              somma. </s>
              <s>Se perciò si chiamino
                <emph type="italics"/>
              s.r
                <emph.end type="italics"/>
              i seni retti,
                <emph type="italics"/>
              e
                <emph.end type="italics"/>
              le corde, e con Ŗ si signi­
                <lb/>
              fichi la loro somma, avremo 2Ŗ
                <emph type="italics"/>
              s.r
                <emph.end type="italics"/>
              =Ŗc. </s>
              <s>E se con Ŗ
                <emph type="italics"/>
              r
                <emph.end type="italics"/>
              si rappresenti la
                <lb/>
              somma dei raggi, sarà, per il terzo premesso principio, Ŗ
                <emph type="italics"/>
              c
                <emph.end type="italics"/>
              :Ŗ
                <emph type="italics"/>
              r
                <emph.end type="italics"/>
              =AF:IMF=
                <lb/>
              2Ŗ
                <emph type="italics"/>
              s.r
                <emph.end type="italics"/>
              :Ŗ
                <emph type="italics"/>
              r.
                <emph.end type="italics"/>
              E perchè, per il primo degli stessi premessi principii, Ŗ
                <emph type="italics"/>
              s.r
                <emph.end type="italics"/>
              :Ŗ
                <emph type="italics"/>
              r
                <emph.end type="italics"/>
              =
                <lb/>
              IQ:IM, ossia 2Ŗ
                <emph type="italics"/>
              s.r
                <emph.end type="italics"/>
              :Ŗ
                <emph type="italics"/>
              r
                <emph.end type="italics"/>
              =2IQ:IM; dunque AF:IMF=2IQ:IM=
                <lb/>
              4IQ:2IM=4IQ:IMF, ond'è veramente AF=4Iq. </s>
            </p>
            <p type="main">
              <s>Potendosi ora una tale dimostrazione applicare a qualunque punto della
                <lb/>
              mezza Cicloide, comunque sia dall'origine A distante, supponiamo che il dato
                <lb/>
              punto sia D. </s>
              <s>Troveremo ancora, col medesimo processo, AFD=4IL=2IH,
                <lb/>
              ciò che vuol dire essere, così com'era il proposito di dimostrare, la mezza
                <lb/>
              Cicloide doppia al diametro del circolo genitore. </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Corollario.
                <emph.end type="italics"/>
              — Diviso l'arco IR nel mezzo in P, come nel mezzo M è
                <lb/>
              stato diviso l'arco IF, e condotte le due corde FP, RM, è facile vedere che
                <lb/>
              queste s'intersecheranno fra loro e col diametro HI nel punto T, in modo
                <lb/>
              che sia HF=HT=HI—IT=HI—2IQ, d'onde 2HF+4IQ=2HI=
                <lb/>
              AFD, essendo la semicicloide, per le cose già dimostrate, uguale al doppio del
                <lb/>
              diametro. </s>
              <s>E perch'è stato altresì dimostrato che la porzione AF è uguale al
                <lb/>
              quadruplo del seno verso IQ, dunque 2HF=AFD—AF=DF, ciò che
                <lb/>
              vuol dire essere ogni porzione, presa dal vertice, uguale al doppio della tan­
                <lb/>
              gente. </s>
              <s>Così il Wallis, quell'
                <emph type="italics"/>
              Anglus vir doctissimus, qui et praelo per se,
                <lb/>
              vel per amicos suo nomine vulgavit
                <emph.end type="italics"/>
              (pag. </s>
              <s>344), formulò la seconda parte
                <lb/>
              della proposiz. </s>
              <s>XXII, nel cap. </s>
              <s>V della sua
                <emph type="italics"/>
              Mechanica:
                <emph.end type="italics"/>
              “ Curvae semicycloi­
                <lb/>
              dis portio quaevis, ad verticem terminata, est dupla subtensae corresponden­
                <lb/>
              tis arcus circuli genitoris ” (Londini 1741, pag. </s>
              <s>424). </s>
            </p>
            <p type="main">
              <s>“ PROPOSITIO II. —
                <emph type="italics"/>
              In rota simplici spatium trochoidis triplum est
                <lb/>
              eiusdem rotae ”
                <emph.end type="italics"/>
              (Ouvr. </s>
              <s>cit., pag. </s>
              <s>310). </s>
            </p>
            <p type="main">
              <s>La facilità della dimostrazione dipende dall'invenzion di quella curva,
                <lb/>
              che il Roberval chiamava la
                <emph type="italics"/>
              Compagne de la roulette,
                <emph.end type="italics"/>
              e noi la
                <emph type="italics"/>
              Comite
                <emph.end type="italics"/>
              della </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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