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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="020/01/2718.jpg" pagenum="343"/>
              quattro quadrati EG: ovvero
                <emph type="italics"/>
              sumptis dimidiis,
                <emph.end type="italics"/>
              come il rettangolo FGH a
                <lb/>
              due quadrati GE.
                <emph type="italics"/>
              Et componendo patet propositum ”
                <emph.end type="italics"/>
              (ivi). </s>
            </p>
            <p type="main">
              <s>Nel
                <emph type="italics"/>
              Raćconto
                <emph.end type="italics"/>
              dei problemi proposti ai Matematici francesi udimmo dianzi
                <lb/>
              il teorema formulato dal Torricelli in altra maniera, alla quale è facile ri­
                <lb/>
              durre questa, ora espressa dalla relazione So:Co=FE2+EG2:2EG2,
                <lb/>
              perch'essendo EG=CB/2, sostituendo, e moltiplicando per
                <foreign lang="grc">π</foreign>
              , avremo So:Co=
                <lb/>
                <foreign lang="grc">π</foreign>
              FE2+
                <foreign lang="grc">π</foreign>
              CB2/4:CB2/2=
                <foreign lang="grc">π</foreign>
              CB2+4
                <foreign lang="grc">π</foreign>
              FE2:2
                <foreign lang="grc">π</foreign>
              CB2, che vuol dire appunto,
                <lb/>
              rammemorandoci che la FE sega l'asse nel mezzo, stare il solido al cono
                <lb/>
              inscritto come una sua base, con quattro medie sezioni, a due basi. </s>
            </p>
            <p type="main">
              <s>Udimmo pure, in quel Racconto, il Torricelli compiacersi di avere in
                <lb/>
              questo suo teorema compendiata una gran parte delle dottrine di Archimede,
                <lb/>
              per conferma di che, specialmente contro i dubitanti della verità delle con­
                <lb/>
              clusioni, alle quali conduceva il metodo del Cavalieri; faceva notare come il
                <lb/>
              detto teorema universalissimo, applicato ai vari casi particolari, concordava
                <lb/>
              con le proposizioni dimostrate ne'libri
                <emph type="italics"/>
              De sphaera et cylindro,
                <emph.end type="italics"/>
              e
                <emph type="italics"/>
              De conoid. </s>
              <s>
                <lb/>
              et sphaeralibus.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>“ Esto conoides parabolicum CFAHD (nella medesima figura 298), conus
                <lb/>
              inscriptus CAD, axis AB sectus bifariam in E, et applicata EF. </s>
              <s>Dixi conoi­
                <lb/>
              des ad conum esse ut duo quadrata ex EF, EG, ad duplum quadrati EG, ut
                <lb/>
              ostensum est. </s>
              <s>Dico convenire cum Archimedis XXIII
                <emph type="italics"/>
              De con. </s>
              <s>et spaer.
                <emph.end type="italics"/>
              Pona­
                <lb/>
              tur enim quadratum EF esse ut duo: erit AD ut quatuor, et ideo EG ut
                <lb/>
              unum. </s>
              <s>Quare, componendo sumptisque consequentium duplis, erit quadratum
                <lb/>
              FE, cum quadrato EG, ad duo quadrata ex EG, ut 3 ad duo. </s>
              <s>” </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              “ Che la proposizione universalissima concordi con quella della Sfera,
                <lb/>
              et con la XXIX De con. </s>
              <s>et spaer.:
                <emph.end type="italics"/>
              Sit hemisphaerium, vel hemisphaeroides
                <lb/>
              ABC (fig. </s>
              <s>211), conus inscriptus ABC, axis BD sectus
                <lb/>
                <figure id="id.020.01.2718.1.jpg" xlink:href="020/01/2718/1.jpg" number="716"/>
              </s>
            </p>
            <p type="caption">
              <s>Figura 211.
                <lb/>
              sit bifariam in E, et applicata EF. </s>
              <s>Dixi hemisphaerium
                <lb/>
              ad conum inscriptum esse ut duo quadrata ex FE et ex
                <lb/>
              EG, ad duplum quadrati EG. </s>
              <s>Probo convenire cum Ar­
                <lb/>
              chimede. </s>
              <s>Esto axis integer BH, ponaturque quadratum
                <lb/>
              FE esse 3. Quadratum FE, ad quadratum AD, est ut
                <lb/>
              rectangulum BFH, ad rectangulum BDH, nempe ut
                <lb/>
              3 ad 4. Quadratum vero AD ad EG est ut 4 ad 1. Ergo
                <lb/>
              ex aequo quadratum FE, ad EG, est ut 3 ad 1. Ergo,
                <lb/>
              componendo, sumptisque consequentium duplis, patet duo quadrata FE, EG,
                <lb/>
              ad duo quadrata EG, esse ut 4 ad 2 ” (MSS. Gal. </s>
              <s>Disc., T. XXX, fol. </s>
              <s>184). </s>
            </p>
            <p type="main">
              <s>Soggiunse il Torricelli a queste due un'altra Nota, per provar
                <emph type="italics"/>
              che la
                <lb/>
              dimostrazione universalissima, nel conoide iperbolico e nella porzion di
                <lb/>
              sferoide, concordi con la volgata di Archimede XXVII e XXXI De conoid. </s>
              <s>
                <lb/>
              et sphaer.
                <emph.end type="italics"/>
              (ivi). Rappresenti AIBC (fig. </s>
              <s>212) una delle iperbole, l'asse DB
                <lb/>
              della quale sia prolungato infino a incontrare in E il vertice dell'altra iper­
                <lb/>
              bola. </s>
              <s>Sia L il centro, ed EO uguale ad EL, cosicchè insomma sia BO ses-</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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