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

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            <p type="main">
              <s>
                <pb xlink:href="020/01/2724.jpg" pagenum="349"/>
              ut quadratum I ad semissem rectanguli EGH, nempe duplus. </s>
              <s>Propterea co­
                <lb/>
              nus, cuius radius basis sit I, altitudo vero MN, aequalis erit sphaeroidi, sive
                <lb/>
              reliquo segmenti conici, demptis duobus conis ” (ibid.). </s>
            </p>
            <p type="main">
              <s>Sia dunque, come vuole il Torricelli, I2=EG.GH. </s>
              <s>Avremo per il
                <lb/>
              lemma primo, significati con
                <emph type="italics"/>
              C
                <emph.end type="italics"/>
              il circolo, e con
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              la lunula,
                <emph type="italics"/>
              C
                <emph.end type="italics"/>
              .I:
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              .EH=
                <lb/>
              I2:EG.GH/2, che vuol dire il circolo esser doppio della lunula, e perciò il
                <lb/>
              cono, la base del quale abbia per raggio I, con l'altezza MN, sarà, per fa­
                <lb/>
              cile corollario dalla XXIX archimedea
                <emph type="italics"/>
              De conoid. </s>
              <s>et sphaer.,
                <emph.end type="italics"/>
              uguale alla
                <lb/>
              sferoide. </s>
            </p>
            <p type="main">
              <s>È il presente proposito quello di dimostrare che una tale sferoide, o il
                <lb/>
              cono a lei equivalente, è medio proporzionale fra i due coni ABD, BDC, le­
                <lb/>
              vati dal frusto, i quali coni, per avere altezza uguale, stanno come i qua­
                <lb/>
              drati de'raggi delle basi, ossia come AN2 a BM2. </s>
              <s>Ma anche il terzo cono, a
                <lb/>
              cui s'è detto uguagliarsi la sferoide, ha la medesima altezza degli altri due;
                <lb/>
              dunque tutto si riduce a dimostrare che il quadrato del raggio I, ossia il
                <lb/>
              rettangolo EG.GH è medio proporzionale tra AN2 e BM2, ciò che si può
                <lb/>
              fare in questa maniera: Abbiamo, per ragion delle parallele, NF:FM=
                <lb/>
              AE:EB. Componendo, NF+FM:FM=AE+EB:EB, ossia NM:FM=
                <lb/>
              AB:EB. </s>
              <s>Ma NM=2FM, dunque AB=2EB, e perciò AD=2EG, ossia
                <lb/>
              AN=EG, come, per le medesime ragioni, GH=BM. </s>
              <s>Ora EG:GH=
                <lb/>
              EG2:EG.GH=EG.GH:GH2, per cui, sostituendo AN2 ad EG2, se ne
                <lb/>
              concluderà il proposito, come il Torricelli stesso lo conclude con questo di­
                <lb/>
              scorso: </s>
            </p>
            <p type="main">
              <s>“ Conum autem praedictum I medium proportionalem esse inter ABD,
                <lb/>
              BDC, patet. </s>
              <s>Nam, cum rectangulum EGH medium sit inter quadratum AN,
                <lb/>
              BM, etiam quadratum I medium erit inter eadem. </s>
              <s>Et propterea conus I, sive
                <lb/>
              sphaerois illa media proportionalis erit inter demptos conos. </s>
              <s>Erit enim, ob
                <lb/>
              parallelas, ut NF ad FM, ita AE ad EB. </s>
              <s>Et componendo etc. </s>
              <s>Sed NM dupla
                <lb/>
              est MF; ergo AB dupla est BE, et propterea AD dupla EG. </s>
              <s>Quare AN et
                <lb/>
              EG sunt aequales, et GH, BM sunt aequales. </s>
              <s>Quadratum vero EG, ad rectan­
                <lb/>
              gulum EGG, est ut EG ad GH, et rectangulum EGH, ad quadratum GH,
                <lb/>
              est ut EG ad GH. </s>
              <s>Quare patet propositum ” (ibid.). </s>
            </p>
            <p type="main">
              <s>Così dimostrava il Torricelli, con la fecondità del suo proprio ingegno,
                <lb/>
              in una maniera forse diversa da quelle tre usate dal Ricci, la risoluzione del
                <lb/>
              frusto conico in tre coni di altezze uguali. </s>
              <s>Se non che al terzo cono di mezzo
                <lb/>
              sostituiva una sferoide, perchè l'intento suo principale era quello di trasporre
                <lb/>
              la bella proposizione, dal campo della Stereometria pura, dove lo stesso Ricci
                <lb/>
              l'aveva lasciata, in quello della Baricentrica. </s>
              <s>Riducendosi infatti il centro di
                <lb/>
              gravità di essa sferoide nel mezzo dell'asse, si venivano a render più sem­
                <lb/>
              plici, nella libbra gravata delle parti, nelle quali era il solido risoluto, le ra­
                <lb/>
              gioni delle equiponderanze. </s>
            </p>
            <p type="main">
              <s>Venne al nostro Autore l'occasione di far ciò, essendo intorno a esami­
                <lb/>
              nare le proposizioni galileiane
                <emph type="italics"/>
              De centro gravitatis,
                <emph.end type="italics"/>
              alcuna delle quali essen-</s>
            </p>
          </chap>
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      </text>
    </archimedes>
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