Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[31. LEMMAI.]
[32. LEMMA II.]
[33. LEMMA III.]
[34. LEMMA IIII.]
[35. PROPOSITIO VII.]
[36. PROPOSITIO VIII.]
[37. COMMENTARIVS.]
[38. PROPOSITIO IX.]
[39. COMMENTARIVS.]
[40. PROPOSITIO X.]
[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.]
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ARCHIMEDIS
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        <div type="section" level="1" n="41">
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            <s xml:space="preserve">
              <pb file="0072" n="72" rhead="ARCHIMEDIS"/>
            quindecim ad quatuor; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ad eam, quæ uſque ad axem maiorem pro
              <lb/>
            portionem habeat: </s>
            <s xml:space="preserve">erit quæ uſ que ad axem minor ipſa k c.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="6">
            <note position="right" xlink:label="note-0071-07" xlink:href="note-0071-07a" xml:space="preserve">F</note>
          </div>
          <note position="left" xml:space="preserve">10. quinti</note>
          <p>
            <s xml:space="preserve">Sit ei, quæ uſque ad axem æ qualis k r.</s>
            <s xml:space="preserve">] _Hac nos addidimus,_
              <lb/>
              <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/>
            _quæ in translatione non erant._</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="7">
            <note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">G</note>
          </div>
          <p style="it">
            <s xml:space="preserve">_Eſt autem & </s>
            <s xml:space="preserve">s b ſeſquialtera ipſius b r.</s>
            <s xml:space="preserve">]_ Ponitur enim
              <lb/>
              <anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/>
            d b ſeſquialtera ipſius b k; </s>
            <s xml:space="preserve">itémq; </s>
            <s xml:space="preserve">d ſ ſeſquialtera k r. </s>
            <s 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:space="preserve">ergo & </s>
            <s xml:space="preserve">reliqua
              <lb/>
              <anchor type="note" xlink:label="note-0072-04a" xlink:href="note-0072-04"/>
            s b ad reliquim b r, ut d b ad b k.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="8">
            <note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">H</note>
            <note position="left" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">19. quinti</note>
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          <p style="it">
            <s xml:space="preserve">_Quæ ſimiles ſint portioni a b l.</s>
            <s xml:space="preserve">]_ Similes portiones coni ſe-
              <lb/>
              <anchor type="note" xlink:label="note-0072-05a" xlink:href="note-0072-05"/>
            ctionum Apollonius it. </s>
            <s 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:space="preserve">hoc est. </s>
            <s xml:space="preserve">in quibus ſi du-
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            cantnr lineæ æquidistantes baſi numero æquales: </s>
            <s xml:space="preserve">æquidiſtantes atq;
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            </s>
            <s xml:space="preserve">baſes ad partes diametrorum, quæ ab ipſis ad uerticem abſcindũtur,
              <lb/>
            eandem proportionem babent: </s>
            <s xml:space="preserve">it émq; </s>
            <s xml:space="preserve">partes abſciſſæ ad abſciſſas. </s>
            <s xml:space="preserve">
              <lb/>
            ducuntur autem lineæ baſi æquidistantes: </s>
            <s xml:space="preserve">ut opinor, deſcripta in ſin
              <lb/>
            gulis plane rectilinea figura, quæ lateribus numero æqualibus conti
              <lb/>
              <anchor type="note" xlink:label="note-0072-06a" xlink:href="note-0072-06"/>
            neatur. </s>
            <s xml:space="preserve">Itaq; </s>
            <s xml:space="preserve">portiones ſimiles à ſimilibus coni ſectionibus abſcindũ
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            tur: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">earum diametri ſiue ad baſes rectæ, ſiue cum baſibus æ qua-
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            les angulos facientes, ad ipſas baſes eandem habent proportionem.</s>
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          </p>
          <div type="float" level="2" n="9">
            <note position="left" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">K</note>
            <note position="left" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">γνωρίμως</note>
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          <p style="it">
            <s xml:space="preserve">_Tranſibit igitur a e i coni ſectio per k.</s>
            <s xml:space="preserve">]_ Sienim fieri po
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              <anchor type="note" xlink:label="note-0072-07a" xlink:href="note-0072-07"/>
            teſt non tranſeat per k, ſed per aliud punctum lineæ d b, ut per u.
              <lb/>
            </s>
            <s xml:space="preserve">Quoniam igitur in rectáguli coni ſectione a e i, cuius diameter e z,
              <lb/>
            ducta eſt a e, & </s>
            <s xml:space="preserve">producta: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">d b diametro æquidistans utraſque
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            a e, a i ſecat; </s>
            <s xml:space="preserve">a e quidem in b, ai uero in d: </s>
            <s 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:space="preserve">Archimedis de quadratura parabol
              <unsure/>
            æ. </s>
            <s xml:space="preserve">Sed a z ſeſquialtera eſt
              <lb/>
            ipſius z d: </s>
            <s xml:space="preserve">eſt enim ut tria ad duo, quod mox demonſtrabimus. </s>
            <s xml:space="preserve">ergo
              <lb/>
            d b ſeſquialtera eſt ipſius b u. </s>
            <s xml:space="preserve">eſt auté d b & </s>
            <s xml:space="preserve">ipſius b k ſeſquialte
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            ra. </s>
            <s xml:space="preserve">quare lineæ b u, b k inter ſe æ quales ſunt; </s>
            <s xml:space="preserve">quod fieri non po-
              <lb/>
              <anchor type="note" xlink:label="note-0072-08a" xlink:href="note-0072-08"/>
            teſt. </s>
            <s xml:space="preserve">restanguli igitur com ſectio a e i per punctum k tranſibit.
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            </s>
            <s xml:space="preserve">quod demonstrare uolebamus.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="10">
            <note position="left" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">L</note>
            <note position="left" xlink:label="note-0072-08" xlink:href="note-0072-08a" xml:space="preserve">2. quinti.</note>
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