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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="020/01/2759.jpg" pagenum="384"/>
              famiglia delle parabole quadratiche o naturali. </s>
              <s>Intorno a che il Torricelli
                <lb/>
              dimostrò che, se la velocità è quadratica, la parabola che descriverebbe il
                <lb/>
              proietto è cubica: se la velocità è cubica, la parabola è biquadratica, e in
                <lb/>
              generale, se la velocità è di grado
                <emph type="italics"/>
              n,
                <emph.end type="italics"/>
              sarà di grado
                <emph type="italics"/>
              n+1
                <emph.end type="italics"/>
              la potenza della
                <lb/>
              parabola relativa. </s>
              <s>Sebbene sia il concetto assai pellegrino, è nonostante di
                <lb/>
              molto facile dimostrazione, come apparisce dal seguente esempio, applicato al
                <lb/>
              caso della parabola cubica, premessovi questo problema per lemma:
                <lb/>
                <figure id="id.020.01.2759.1.jpg" xlink:href="020/01/2759/1.jpg" number="752"/>
              </s>
            </p>
            <p type="caption">
              <s>Figura 247.</s>
            </p>
            <p type="main">
              <s>“ Si mobile moveatur deorsum tempore AC
                <lb/>
              (fig. </s>
              <s>247), et tempore AB, et augeatur velocitas qua­
                <lb/>
              dratice, quaeritur ratio spatiorum. </s>
              <s>” </s>
            </p>
            <p type="main">
              <s>“ Dico sic: Spatia peracta habent rationem
                <lb/>
              compositam ex ratione velocitatum, et ex ratione
                <lb/>
              temporum. </s>
              <s>Sint spatia peracta AB, AC, tempora
                <lb/>
              vero DE, DF. </s>
              <s>Supponamus mobile in B et in C
                <lb/>
              converti horizontaliter. </s>
              <s>Jam impetus in B, ad im­
                <lb/>
              petum in C, erit ut quadratum temporis DE, ad
                <lb/>
              quadratum DF. </s>
              <s>Ergo spatium BH, factum tempore casus AB, ad spatium CI,
                <lb/>
              factum tempore casus AC, rationem habebit compositam rectae DE ad DF,
                <lb/>
              et quadrati DE ad quadratum DF. </s>
              <s>Ergo spatium BH ad CI erit ut cubus DE
                <lb/>
              ad DF. </s>
              <s>Sed ut spatia BH, CI, ita sunt spatia AB, AC, ipsorum submultiplicia
                <lb/>
              aequaliter, ergo patet etc. </s>
              <s>” </s>
            </p>
            <p type="main">
              <s>“ PROPOSIZIONE XII. —
                <emph type="italics"/>
              Cadat mobile aliquod horizontaliter concitatum
                <lb/>
              ex plano DA
                <emph.end type="italics"/>
              (fig. </s>
              <s>248),
                <emph type="italics"/>
              ita ut duos impetus habeat, alterum aequabilem
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.020.01.2759.2.jpg" xlink:href="020/01/2759/2.jpg" number="753"/>
              </s>
            </p>
            <p type="caption">
              <s>Figura 248.
                <lb/>
                <emph type="italics"/>
              horizontalem versus partes EC. alterum de­
                <lb/>
              scendentem acceleratum quadratice. </s>
              <s>Dico pa­
                <lb/>
              rabolam cubicam fieri. </s>
              <s>”
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>“ Hoc ex dictis patet. </s>
              <s>Nam consideretur
                <lb/>
              mobile in quibuslibet punctis B, C. </s>
              <s>Cum im­
                <lb/>
              petus horizontalis externus sit et aequabilis,
                <lb/>
              erunt CI, BH ut tempora casuum. </s>
              <s>Sed spatia
                <lb/>
              peracta EC, FB sunt ut cubi temporum; ergo
                <lb/>
              cubi rectarum CI, BH erunt ut EC, FB, sive
                <lb/>
              ut IA ad AH ” (ibid., T. XXXI, fol. </s>
              <s>341). </s>
            </p>
            <p type="main">
              <s>Perchè dunque la proposta verità, dato il lemma, è patente, si può quello
                <lb/>
              stesso lemma dimostrare nella sua universalità, d'onde ne derivi la univer­
                <lb/>
              salità sua anche la proposizione ora scritta. </s>
              <s>Chiamati S, S′, V, V′, T, T′ due
                <lb/>
              spazi, due varie velocità, due vari tempi, abbiamo, per le note leggi del moto,
                <lb/>
              S:S′=V.T:V′.T′. </s>
              <s>Che se l'accelerazione è lineare, ossia se V:V′=
                <lb/>
              T:T′, sarà S:S′=T2:T′2; se l'accelerazione è quadratica, e perciò V:V=
                <lb/>
              T2:T′2, sarà S:S′=T3:T′3: se poi l'accelerazione è cubica, e V:V′=
                <lb/>
              T3:T′3, sarà S:S′=T1:T′1, e in generale, se l'accelerazione è di grado
                <emph type="italics"/>
              n,
                <emph.end type="italics"/>
                <lb/>
              sarà S:S′=T
                <emph type="italics"/>
              n+1
                <emph.end type="italics"/>
                <emph type="italics"/>
              n+1
                <emph.end type="italics"/>
              .</s>
              <s>Cosicchè, facendone l'applicazione alla para­
                <lb/>
              bola, rappresentata dalla stessa ultima figura, sarà l'equazione di lei espressa
                <lb/>
              da AH:AI=HB</s>
              <s>
                <emph type="italics"/>
              n+1
                <emph.end type="italics"/>
              :IC
                <emph type="italics"/>
              n+1
                <emph.end type="italics"/>
              .</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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