Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[91.] THEOREMA XXI. PROPOSITIO XXVI.
[92.] THEOREMA XXII. PROPOSITIO XXVII.
[93.] PROBLEMA VI. PROPOSITIO XX VIII.
[94.] THE OREMA XXIII. PROPOSITIO XXIX.
[95.] THEOREMA XXIIII. PROPOSITIO XXX.
[96.] THEOREMA XXV. PROPOSITIO XXXI.
[97.] FINIS LIBRI DE CENTRO GRAVITATIS SOLIDORVM.
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          <p style="it">
            <s xml:id="echoid-s1761" xml:space="preserve">
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            quindecim ad quatuor; </s>
            <s xml:id="echoid-s1762" xml:space="preserve">& </s>
            <s xml:id="echoid-s1763" xml:space="preserve">ad eam, quæ uſque ad axem maiorem pro
              <lb/>
            portionem habeat: </s>
            <s xml:id="echoid-s1764" xml:space="preserve">erit quæ uſ que ad axem minor ipſa k c.</s>
            <s xml:id="echoid-s1765" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">10. quinti</note>
          <p>
            <s xml:id="echoid-s1766" xml:space="preserve">Sit ei, quæ uſque ad axem æ qualis k r.</s>
            <s xml:id="echoid-s1767" xml:space="preserve">] _Hac nos addidimus,_
              <lb/>
              <note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">G</note>
            _quæ in translatione non erant._</s>
            <s xml:id="echoid-s1768" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1769" xml:space="preserve">_Eſt autem & </s>
            <s xml:id="echoid-s1770" xml:space="preserve">s b ſeſquialtera ipſius b r.</s>
            <s xml:id="echoid-s1771" xml:space="preserve">]_ Ponitur enim
              <lb/>
              <note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">H</note>
            d b ſeſquialtera ipſius b k; </s>
            <s xml:id="echoid-s1772" xml:space="preserve">itémq; </s>
            <s xml:id="echoid-s1773" xml:space="preserve">d ſ ſeſquialtera k r. </s>
            <s xml:id="echoid-s1774" xml:space="preserve">quare ut to
              <lb/>
            ta d b ad totam b K, ita pars d s ad partem K r. </s>
            <s xml:id="echoid-s1775" xml:space="preserve">ergo & </s>
            <s xml:id="echoid-s1776" xml:space="preserve">reliqua
              <lb/>
              <note position="left" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">19. quinti</note>
            s b ad reliquim b r, ut d b ad b k.</s>
            <s xml:id="echoid-s1777" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1778" xml:space="preserve">_Quæ ſimiles ſint portioni a b l.</s>
            <s xml:id="echoid-s1779" xml:space="preserve">]_ Similes portiones coni ſe-
              <lb/>
              <note position="left" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">K</note>
            ctionum Apollonius it. </s>
            <s xml:id="echoid-s1780" xml:space="preserve">i diffiniuit in ſexto libro conicorum, ut ſcri-
              <lb/>
            bit Eutocius, εν οἱς α χ θεισω
              <unsure/>
            νἐν ἑηάστω παραλλήλων τῆ βάσει, ἵσωι
              <lb/>
            τὸ πλῆθος, ὰι παρὰλληλοι, καὶ αἱ βάσ{ει}ς πρ
              <unsure/>
            ὸς τὰςἀποτεμνομένας
              <lb/>
            ἀπὸ τῶν διαμέ τρων ταῖς νορυφαῖς ἐν τοῖς αὐτοῖ ς λὄγοιςεἰσἰ, καὶἁι
              <lb/>
            ἀποτεμνόμεναι πρ
              <unsure/>
            ὸς τάς ἀποτεμνομένας; </s>
            <s xml:id="echoid-s1781" xml:space="preserve">hoc est. </s>
            <s xml:id="echoid-s1782" xml:space="preserve">in quibus ſi du-
              <lb/>
            cantnr lineæ æquidistantes baſi numero æquales: </s>
            <s xml:id="echoid-s1783" xml:space="preserve">æquidiſtantes atq;
              <lb/>
            </s>
            <s xml:id="echoid-s1784" xml:space="preserve">baſes ad partes diametrorum, quæ ab ipſis ad uerticem abſcindũtur,
              <lb/>
            eandem proportionem babent: </s>
            <s xml:id="echoid-s1785" xml:space="preserve">it émq; </s>
            <s xml:id="echoid-s1786" xml:space="preserve">partes abſciſſæ ad abſciſſas. </s>
            <s xml:id="echoid-s1787" xml:space="preserve">
              <lb/>
            ducuntur autem lineæ baſi æquidistantes: </s>
            <s xml:id="echoid-s1788" xml:space="preserve">ut opinor, deſcripta in ſin
              <lb/>
            gulis plane rectilinea figura, quæ lateribus numero æqualibus conti
              <lb/>
              <note position="left" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">γνωρίμως</note>
            neatur. </s>
            <s xml:id="echoid-s1789" xml:space="preserve">Itaq; </s>
            <s xml:id="echoid-s1790" xml:space="preserve">portiones ſimiles à ſimilibus coni ſectionibus abſcindũ
              <lb/>
            tur: </s>
            <s xml:id="echoid-s1791" xml:space="preserve">& </s>
            <s xml:id="echoid-s1792" xml:space="preserve">earum diametri ſiue ad baſes rectæ, ſiue cum baſibus æ qua-
              <lb/>
            les angulos facientes, ad ipſas baſes eandem habent proportionem.</s>
            <s xml:id="echoid-s1793" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1794" xml:space="preserve">_Tranſibit igitur a e i coni ſectio per k.</s>
            <s xml:id="echoid-s1795" xml:space="preserve">]_ Sienim fieri po
              <lb/>
              <note position="left" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">L</note>
            teſt non tranſeat per k, ſed per aliud punctum lineæ d b, ut per u.
              <lb/>
            </s>
            <s xml:id="echoid-s1796" xml:space="preserve">Quoniam igitur in rectáguli coni ſectione a e i, cuius diameter e z,
              <lb/>
            ducta eſt a e, & </s>
            <s xml:id="echoid-s1797" xml:space="preserve">producta: </s>
            <s xml:id="echoid-s1798" xml:space="preserve">& </s>
            <s xml:id="echoid-s1799" xml:space="preserve">d b diametro æquidistans utraſque
              <lb/>
            a e, a i ſecat; </s>
            <s xml:id="echoid-s1800" xml:space="preserve">a e quidem in b, ai uero in d: </s>
            <s xml:id="echoid-s1801" xml:space="preserve">habebit d b ad b u
              <lb/>
            proportionem eandem, quam a z, ad z d, ex quarta propoſitione li
              <lb/>
            bri. </s>
            <s xml:id="echoid-s1802" xml:space="preserve">Archimedis de quadratura parabol
              <unsure/>
            æ. </s>
            <s xml:id="echoid-s1803" xml:space="preserve">Sed a z ſeſquialtera eſt
              <lb/>
            ipſius z d: </s>
            <s xml:id="echoid-s1804" xml:space="preserve">eſt enim ut tria ad duo, quod mox demonſtrabimus. </s>
            <s xml:id="echoid-s1805" xml:space="preserve">ergo
              <lb/>
            d b ſeſquialtera eſt ipſius b u. </s>
            <s xml:id="echoid-s1806" xml:space="preserve">eſt auté d b & </s>
            <s xml:id="echoid-s1807" xml:space="preserve">ipſius b k ſeſquialte
              <lb/>
            ra. </s>
            <s xml:id="echoid-s1808" xml:space="preserve">quare lineæ b u, b k inter ſe æ quales ſunt; </s>
            <s xml:id="echoid-s1809" xml:space="preserve">quod fieri non po-
              <lb/>
              <note position="left" xlink:label="note-0072-08" xlink:href="note-0072-08a" xml:space="preserve">2. quinti.</note>
            teſt. </s>
            <s xml:id="echoid-s1810" xml:space="preserve">restanguli igitur com ſectio a e i per punctum k tranſibit.
              <lb/>
            </s>
            <s xml:id="echoid-s1811" xml:space="preserve">quod demonstrare uolebamus.</s>
            <s xml:id="echoid-s1812" xml:space="preserve"/>
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