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

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            <p type="main">
              <s>
                <pb xlink:href="020/01/2063.jpg" pagenum="306"/>
              zio AB passato nel primo. </s>
              <s>Con un ragionamento simile seguitava a dimo­
                <lb/>
              strar Galileo che EK, KR, spazi passati dal mobile nel IIIo e nel IVo tempo,
                <lb/>
              erano cinque e sette volte più grandi dello spazio AB, cosicchè ne conclu­
                <lb/>
              deva che i ricercati eccessi stavano come la serie de'numeri impari 3, 5, 7....
                <lb/>
              E giacchè numerati, nella linea della caduta AR, gli spazi ordinatamente ai
                <lb/>
              tempi, si vede che, se alla fine del Io lo spazio è 1, alla fine del IIo è 4,
                <lb/>
              del IIIo è 9, del IVo è sedici; si confermava per la nuova dimostrazione quel
                <lb/>
              ch'era riuscito Galileo a dimostrare per altre vie, che cioè crescono gli
                <lb/>
              spazi come i quadrati dei tempi. </s>
            </p>
            <p type="main">
              <s>Nel 1622 il Cavalieri propose, come altrove vedemmo, il suo metodo
                <lb/>
              degl'indivisibili a Galileo, il quale lo trovò opportunissimo a rendere anche
                <lb/>
              più perfette queste dimostrazioni per via geometrica, facendo rappresentare
                <lb/>
              gl'infiniti istanti, contenuti in un tempo quanto, agl'infiniti punti contenuti
                <lb/>
              in una linea, e gli spazi alle infinite linee di che si contenesse, secondo il Ca­
                <lb/>
              valieri, una superfice. </s>
              <s>Perciò al Sagredo che, servendosi di numeri deter­
                <lb/>
              minati, avea concluso il sopra riferito ragionamento, il Salviati soggiungeva:
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              “ Voi mi avete fatto venire in mente di aggiungere qualche cosa di più,
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              imperocchè, essendo nel moto accelerato l'agumento continuo, non si pos­
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              sono compartire i gradi della velocità, la quale sempre cresce, in numero
                <lb/>
              alcuno determinato, perchè mutandosi di momento in momento son sempre
                <lb/>
              infiniti: però meglio potremo esemplificare la nostra intenzione, figurandoci
                <lb/>
              un triangolo ” (Alb. </s>
              <s>I, 251, 52). </s>
            </p>
            <p type="main">
              <s>La dimostrazione delle proprietà dei moti accelerati riusciva, per que­
                <lb/>
              sta nuova via geometrica, di una facilità e di un'evidenza maravigliosa, im­
                <lb/>
                <figure id="id.020.01.2063.1.jpg" xlink:href="020/01/2063/1.jpg" number="333"/>
              </s>
            </p>
            <p type="caption">
              <s>Figura 142.
                <lb/>
              perocchè, figurandoci essere quel triangolo AFH (fig. </s>
              <s>142)
                <lb/>
              si può immaginare che le parti uguali AC, CD, DE, EF,
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              prese sopra il lato AF perpendicolare, rappresentino i tempi,
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              e che le linee CG, DK, EI, FH, orizzontalmente condotte
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              parallele alla base FH, rappresentino le velocità via via
                <lb/>
              crescenti, per le proprietà dei triangoli simili, a propor­
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              zione dei tempi. </s>
              <s>Gli spazi perciò, che si sa avere la ragion
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              composta delle velocità e dei tempi, saranno rappresentati
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              dai triangoli ACG, ADK, AEI, AFH, aventi AC, AD, AE,
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              AF per altezze, e CG, DK, EI, FH per loro respettive basi;
                <lb/>
              per cui, chiamandosi per brevità S, S′, S″ quegli stessi spazi,
                <lb/>
              o i triangoli a cui sono proporzionali, sarà S:S′:S″=
                <lb/>
              ACXCG:ADXDK:AEXEI, e perchè AC:AD:AE=
                <lb/>
              CG:DK:EI, dunque S:S′:S″=AC2:AD2:AE2, ossia gli spazi stanno
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              come i quadrati dei tempi. </s>
              <s>Dai trapezi inoltre CK, DI, EH, che si potranno
                <lb/>
              significare per brevità con T, T′, T″, verranno rappresentati gl'incrementi
                <lb/>
              degli spazi via via decorsi, e perchè T=3CGXCD/2, T′=5CGXDE/2,
                <lb/>
              T″=7CGXEF/2, e perciò T:T′:T″....=3:5:7.... </s>
            </p>
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