Ghetaldi, Marino, Marini Ghetaldi Promotvs Archimedis sev de varijs corporum generibus grauitate & magnitudine comparatis

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              <pb o="4" file="0016" n="16" rhead="PROMOTVS"/>
            quam A, primus terminus in ſerie ſecunda ad BC, ſecundum, ergo, & </s>
            <s xml:id="echoid-s174" xml:space="preserve">
              <lb/>
            D, tertius terminus in ſerie prima ad EI, quartum, minorem habebit
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            rationem quam D, tertius terminus in ſerie ſecunda ad EG, quartum.
              <lb/>
            </s>
            <s xml:id="echoid-s175" xml:space="preserve">Quoniam igitur D, minorem habet rationem ad EI, quam ad EG,
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            erit EI, maior quam EG, quod eſt abſurdum. </s>
            <s xml:id="echoid-s176" xml:space="preserve">Non igitur eſt vt
              <note position="left" xlink:label="note-0016-01" xlink:href="note-0016-01a" xml:space="preserve">10. 5.
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              Elem.</note>
            ad BC, ita D, ad maiorem quam EF, oſtenſum autem eſt neque ad mi-
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            norem; </s>
            <s xml:id="echoid-s177" xml:space="preserve">quare vt A, ad BC, ita erit D, ad EF. </s>
            <s xml:id="echoid-s178" xml:space="preserve">& </s>
            <s xml:id="echoid-s179" xml:space="preserve">incommenſurabilia igi-
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            tur corpora eiuſdem generis eandem in grauitate rationem habent,
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            quam in magnitudine, quod erat demonſtrandum.</s>
            <s xml:id="echoid-s180" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s181" xml:space="preserve">ID QVOD nos duobus præcedentibus Theorematis de-
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            monſtrauimus, nõnulli, vt per ſe notum, & </s>
            <s xml:id="echoid-s182" xml:space="preserve">vt commune quod-
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            dam axioma ſupponunt, quam bene & </s>
            <s xml:id="echoid-s183" xml:space="preserve">rationabiliter ipſi vide-
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            rint; </s>
            <s xml:id="echoid-s184" xml:space="preserve">melius enim Euclides propoſitionem 20. </s>
            <s xml:id="echoid-s185" xml:space="preserve">primi libri Ele-
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            mentorum ſuppoſuiſſet vt pronunciatum; </s>
            <s xml:id="echoid-s186" xml:space="preserve">vnicuique enim no-
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            tius eſt duo trianguli latera reliquo eſſe maiora (cum & </s>
            <s xml:id="echoid-s187" xml:space="preserve">Aſino
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            illud ſit notum) quam corpora grauia eiuſdem generis eandem
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            in grauitate rationem habere, quam in magnitudine, & </s>
            <s xml:id="echoid-s188" xml:space="preserve">tamen
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            illam propoſitionem demonſtrat Euclides, non ſupponit, non
              <lb/>
            igitur hæc, quæ minus ad principij rationem accedit, ſuppo-
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            nenda fuit, ſed demonſtranda.</s>
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          <head xml:id="echoid-head12" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head>
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            <s xml:id="echoid-s190" xml:space="preserve">SI quatuor corporum grauium primum ad ſecundũ
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            eandem in magnitudine rationem habeat, quam
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            tertium ad quartum, primum autem, & </s>
            <s xml:id="echoid-s191" xml:space="preserve">ſecundum ſint
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            eiuſdem generis, itidem tertium, & </s>
            <s xml:id="echoid-s192" xml:space="preserve">quartum; </s>
            <s xml:id="echoid-s193" xml:space="preserve">& </s>
            <s xml:id="echoid-s194" xml:space="preserve">in gra-
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            uitate primum ad ſecundum eandem rationem habebit,
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            quam tertium ad quartum.</s>
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          </p>
          <p>
            <s xml:id="echoid-s196" xml:space="preserve">PRIMVM enim A, ad ſecundum B, eandem in magnitudine ra-
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            tionem habeat, quam tertium C, ad quartum D, ſint autem A, B,
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            eiuſdem generis, itidem C, D. </s>
            <s xml:id="echoid-s197" xml:space="preserve">Dico & </s>
            <s xml:id="echoid-s198" xml:space="preserve">in grauitate primum A, ad
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            ſecundum B, eandem rationem habere, quam tertium C, ad D, quar-
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            tum. </s>
            <s xml:id="echoid-s199" xml:space="preserve">Sint enim earum grauitates E, ipſius A, & </s>
            <s xml:id="echoid-s200" xml:space="preserve">F, ipſius B, ipſius
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            vero C, ſit grauitas G, & </s>
            <s xml:id="echoid-s201" xml:space="preserve">ipſius D, grauitas H, quoniam igitur </s>
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