Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[41.] COMMENTARIVS.
[42.] LEMMA I.
[43.] LEMMA II.
[44.] LEMMA III.
[45.] LEMMA IIII.
[46.] LEMMA V.
[47.] LEMMA VI.
[48.] II.
[49.] III.
[50.] IIII.
[51.] V.
[52.] DEMONSTRATIO SECVNDAE PARTIS.
[53.] COMMENTARIVS.
[54.] DEMONSTRATIO TERTIAE PARTIS.
[55.] COMMENTARIVS.
[56.] DEMONSTRATIO QVARTAE PARTIS.
[57.] DEMONSTRATIO QVINT AE PARTIS.
[58.] FINIS LIBRORVM ARCHIMEDIS DE IIS, QVAE IN AQVA VEHVNTVR.
[59.] FEDERICI COMMANDINI VRBINATIS LIBER DE CENTRO GRAVITATIS SOLIDORV M.
[60.] CVM PRIVILEGIO IN ANNOS X. BONONIAE, Ex Officina Alexandri Benacii. M D LXV.
[61.] ALEXANDRO FARNESIO CARDINALI AMPLISSIMO ET OPTIMO.
[62.] FEDERICI COMMANDINI VRBINATIS LIBER DE CENTRO GRAVITATIS SOLIDORVM. DIFFINITIONES.
[63.] PETITIONES.
[64.] THEOREMA I. PROPOSITIO I.
[65.] THEOREMA II. PROPOSITIO II.
[66.] THE OREMA III. PROPOSITIO III.
[67.] THE OREMA IIII. PROPOSITIO IIII.
[68.] ALITER.
[69.] THEOREMA V. PROPOSITIO V.
[70.] COROLLARIVM.
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              <pb o="36" file="0183" n="183" rhead="DE CENTRO GRAVIT. SOLID."/>
            grauitatis magnitudinis, quæ ex utriſque pyramidibus cõ
              <lb/>
            ſtat; </s>
            <s xml:id="echoid-s4564" xml:space="preserve">hoc eſt ipſius fruſti. </s>
            <s xml:id="echoid-s4565" xml:space="preserve">Sed fruſti centrum eſt etiam in a-
              <lb/>
            xe g h. </s>
            <s xml:id="echoid-s4566" xml:space="preserve">ergo in puncto φ, in quo lineæ z u, g h conueniunt.
              <lb/>
            </s>
            <s xml:id="echoid-s4567" xml:space="preserve">Itaque u φ ad φ z eam proportionem habet, quam pyramis
              <lb/>
              <note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">8. prim I
                <lb/>
              libri Ar-
                <lb/>
              chimedis
                <lb/>
              de cẽtro
                <lb/>
              grauita-
                <lb/>
              tis plano
                <lb/>
              runi</note>
            b c f e d ad pyramidem a b c d. </s>
            <s xml:id="echoid-s4568" xml:space="preserve">& </s>
            <s xml:id="echoid-s4569" xml:space="preserve">componendo u z ad z φ
              <lb/>
            eam habet, quam fruſtum ad pyramidem a b c d. </s>
            <s xml:id="echoid-s4570" xml:space="preserve">Vtuero
              <lb/>
            u z ad z φ, ita o p ad p φ ob ſimilitudinem triangulorum,
              <lb/>
            u o φ, z p φ. </s>
            <s xml:id="echoid-s4571" xml:space="preserve">quare o p ad p φ eſt ut fruſtum ad pyramidem
              <lb/>
            a b c d. </s>
            <s xml:id="echoid-s4572" xml:space="preserve">ſed ita erat o p ad p q. </s>
            <s xml:id="echoid-s4573" xml:space="preserve">æquales igitur ſunt p φ, p q: </s>
            <s xml:id="echoid-s4574" xml:space="preserve">& </s>
            <s xml:id="echoid-s4575" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">7. quinti.</note>
            q φ unum atque idem punctum. </s>
            <s xml:id="echoid-s4576" xml:space="preserve">ex quibus ſequitur lineam
              <lb/>
            z u ſecare o p in q: </s>
            <s xml:id="echoid-s4577" xml:space="preserve">& </s>
            <s xml:id="echoid-s4578" xml:space="preserve">propterea pũctum q ipſius fruſti gra-
              <lb/>
            uitatis centrum eſſe.</s>
            <s xml:id="echoid-s4579" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4580" xml:space="preserve">Sit fruſtum a g à pyramide, quæ quadrangularem baſim
              <lb/>
            habeat abſciſſum, cuius maior baſis a b c d, minor e f g h,
              <lb/>
            & </s>
            <s xml:id="echoid-s4581" xml:space="preserve">axis k l. </s>
            <s xml:id="echoid-s4582" xml:space="preserve">diuidatur autem primũ _k_ l, ita ut quam propor-
              <lb/>
            tionem habet duplum lateris a b unà cum latere e f ad du
              <lb/>
            plum lateris e f unà cum a b; </s>
            <s xml:id="echoid-s4583" xml:space="preserve">habeat k m ad m l. </s>
            <s xml:id="echoid-s4584" xml:space="preserve">deinde à
              <lb/>
            púcto m ad k ſumatur quarta pars ipſius m k, quæ ſit m n.
              <lb/>
            </s>
            <s xml:id="echoid-s4585" xml:space="preserve">& </s>
            <s xml:id="echoid-s4586" xml:space="preserve">rurſus ab l ſumatur quarta pars totius axis l k, quæ ſit
              <lb/>
            l o. </s>
            <s xml:id="echoid-s4587" xml:space="preserve">poſtremo fiat o n ad n p, ut fruſtum a g ad pyramidẽ,
              <lb/>
            cuius baſis ſit eadem, quæ fruſti, & </s>
            <s xml:id="echoid-s4588" xml:space="preserve">altitudo æqualis. </s>
            <s xml:id="echoid-s4589" xml:space="preserve">Dico
              <lb/>
            punctum p fruſti a g grauitatis centrum eſſe. </s>
            <s xml:id="echoid-s4590" xml:space="preserve">ducantur
              <lb/>
            enim a c, e g: </s>
            <s xml:id="echoid-s4591" xml:space="preserve">& </s>
            <s xml:id="echoid-s4592" xml:space="preserve">intelligantur duo fruſta triangulares ba-
              <lb/>
            ſes habentia, quorum alterum l f ex baſibus a b c, e f g cõ-
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            ſtet; </s>
            <s xml:id="echoid-s4593" xml:space="preserve">alterum l h ex baſibus a c d, e g h. </s>
            <s xml:id="echoid-s4594" xml:space="preserve">Sitq; </s>
            <s xml:id="echoid-s4595" xml:space="preserve">fruſti l f axis
              <lb/>
            q r; </s>
            <s xml:id="echoid-s4596" xml:space="preserve">in quo grauitatis centrum s: </s>
            <s xml:id="echoid-s4597" xml:space="preserve">fruſti uero l h axis t u, & </s>
            <s xml:id="echoid-s4598" xml:space="preserve">
              <lb/>
            x grauitatis centrum: </s>
            <s xml:id="echoid-s4599" xml:space="preserve">deinde iungantur u r, t q, x s. </s>
            <s xml:id="echoid-s4600" xml:space="preserve">tranſi-
              <lb/>
            bit u r per l: </s>
            <s xml:id="echoid-s4601" xml:space="preserve">quoniam l eſt centrum grauitatis quadran-
              <lb/>
            guli a b c d: </s>
            <s xml:id="echoid-s4602" xml:space="preserve">& </s>
            <s xml:id="echoid-s4603" xml:space="preserve">puncta r u grauitatis centra triangulorum
              <lb/>
            a b c, a c d; </s>
            <s xml:id="echoid-s4604" xml:space="preserve">in quæ quadrangulum ipſum diuiditur. </s>
            <s xml:id="echoid-s4605" xml:space="preserve">eadem
              <lb/>
            quoque ratione t q per punctum _k_ tranſibit. </s>
            <s xml:id="echoid-s4606" xml:space="preserve">At uero pro
              <lb/>
            portiones, ex quibus fruſtorum grauitatis centra inquiri-
              <lb/>
            mus, eædem ſunt in toto ſruſto a g, & </s>
            <s xml:id="echoid-s4607" xml:space="preserve">in fruſtis l f, l h. </s>
            <s xml:id="echoid-s4608" xml:space="preserve">Sunt
              <lb/>
            enim per octauam huius quadrilatera a b c d, e f g h ſimilia:</s>
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