Theodosius <Bithynius>; Clavius, Christoph, Theodosii Tripolitae Sphaericorum libri tres

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        <div xml:id="echoid-div1094" type="section" level="1" n="539">
          <pb o="401" file="413" n="413" rhead=""/>
          <p>
            <s xml:id="echoid-s13830" xml:space="preserve">SED iam circuli ABCD, AECF, ſecent ſe mutuo ad angulos rectos in
              <lb/>
            punctis A, C; </s>
            <s xml:id="echoid-s13831" xml:space="preserve">ſitq́ue eorum communis ſectio recta AC. </s>
            <s xml:id="echoid-s13832" xml:space="preserve">Diuiſo autem v. </s>
            <s xml:id="echoid-s13833" xml:space="preserve">g.
              <lb/>
            </s>
            <s xml:id="echoid-s13834" xml:space="preserve">ſemicirculo ABC, bifariam in H, vt ſint quadrantes AH, CH, ſumantur
              <lb/>
            duo puncta vtcunque B, G. </s>
            <s xml:id="echoid-s13835" xml:space="preserve">Dico ita eſſe rurſus
              <lb/>
            ſinum arcus AB, ad ſinum arcus, qui per B, ductus
              <lb/>
            rectos angulos facit cum circulo AECF, vt eſt
              <lb/>
              <figure xlink:label="fig-413-01" xlink:href="fig-413-01a" number="261">
                <image file="413-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/413-01"/>
              </figure>
            ſinus arcus AG, ad ſinum arcus, qui per G, ductus
              <lb/>
            cum circulo AECF, rectos facit angulos. </s>
            <s xml:id="echoid-s13836" xml:space="preserve">Quo-
              <lb/>
            niam enim circulus ABC, cum rectus ad circulum
              <lb/>
            AEC, ponatur, tranſit per polos circuli AEC,
              <lb/>
              <note position="right" xlink:label="note-413-01" xlink:href="note-413-01a" xml:space="preserve">13. 1. Theod.
                <lb/>
              Coroll. 16.
                <lb/>
              1. Theod.</note>
            erit H, polus circuli AEC. </s>
            <s xml:id="echoid-s13837" xml:space="preserve">Quare arcus perpen-
              <lb/>
            diculares ad circulum AEC, per puncta B, G, du-
              <lb/>
            cti neceſſario per H, tranſibunt; </s>
            <s xml:id="echoid-s13838" xml:space="preserve">atque adeò arcus
              <lb/>
              <note position="right" xlink:label="note-413-02" xlink:href="note-413-02a" xml:space="preserve">13. 1 Theod.</note>
            illi erunt BA, GA: </s>
            <s xml:id="echoid-s13839" xml:space="preserve">Perſpicuum autem eſt, vt eſt
              <lb/>
            ſinus arcus AB, ad ſinum arcus BA, ita eſſe ſinum
              <lb/>
            arcus AG, ad ſinum arcus GA, cum vtrobique
              <lb/>
            ſit proportro æqualitatis, ſeu identitatis: </s>
            <s xml:id="echoid-s13840" xml:space="preserve">Eſt e-
              <lb/>
            nim idem ſinus arcus AB, & </s>
            <s xml:id="echoid-s13841" xml:space="preserve">arcus BA, necnon
              <lb/>
            idem ſinus arcus AG, & </s>
            <s xml:id="echoid-s13842" xml:space="preserve">arcus GA.</s>
            <s xml:id="echoid-s13843" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13844" xml:space="preserve">QVOD ſi alterum punctorum ſit H, polus circuli AEC, erit quicun-
              <lb/>
            que arcus ex H, ductus, qualis eſt HE, perpendicularis, ad AEC, atque adeò
              <lb/>
              <note position="right" xlink:label="note-413-03" xlink:href="note-413-03a" xml:space="preserve">15. 1 Theod.</note>
            quadrans. </s>
            <s xml:id="echoid-s13845" xml:space="preserve">Rurſus igitur manifeſtum eſt, ita eſſe ſinum arcus AB, ad ſinum ar-
              <lb/>
              <note position="right" xlink:label="note-413-04" xlink:href="note-413-04a" xml:space="preserve">Coroll. 16.
                <lb/>
              1. Theod.</note>
            cus BA, vt eſt ſinus arcus AH, ad ſinum arcus HE, vel HA; </s>
            <s xml:id="echoid-s13846" xml:space="preserve">cum vtrobique
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            quoque ſit æ qualitatis proportio, &</s>
            <s xml:id="echoid-s13847" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13848" xml:space="preserve">Si duo ergo circuli maximi in ſphæra
              <lb/>
            ſe mutuo ſecent, &</s>
            <s xml:id="echoid-s13849" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13850" xml:space="preserve">Quod erat oſtendendum.</s>
            <s xml:id="echoid-s13851" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1100" type="section" level="1" n="540">
          <head xml:id="echoid-head575" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s13852" xml:space="preserve">PERSPICVVM eſt ex demonſtratis: </s>
            <s xml:id="echoid-s13853" xml:space="preserve">Si duo circuli ſe mutuo ſecent, & </s>
            <s xml:id="echoid-s13854" xml:space="preserve">in vno
              <lb/>
            eorum ex duobus punctis vtcunq; </s>
            <s xml:id="echoid-s13855" xml:space="preserve">aſſumptis ducantur ad alterius circuli planum duæ
              <lb/>
            lineæ rectæ perpendiculares; </s>
            <s xml:id="echoid-s13856" xml:space="preserve">ita eſſe ſinum rectum arcus intercepti inter vnum illo-
              <lb/>
            rum punctorum, & </s>
            <s xml:id="echoid-s13857" xml:space="preserve">alterutram circulorum ſectionem, ad perpendicularem ex illo
              <lb/>
            puncto in planum alterius circuli demiſſam, vt eſt ſinus rectus arcus inter alterum
              <lb/>
            punctum, & </s>
            <s xml:id="echoid-s13858" xml:space="preserve">alterutram ſectionem circulorum interiecti, ad perpendicularem ab hoc
              <lb/>
            altero puncto in planum alterius circuli demiſſam. </s>
            <s xml:id="echoid-s13859" xml:space="preserve">Nam in priori figura buius pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s13860" xml:space="preserve">oſtenſum eſt, ita eſſe ſinum arcus
              <emph style="sc">Ab</emph>
            , vel
              <emph style="sc">Cb</emph>
            , ad BP, ſinum rectum arcus BI,
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            vt eſt ſinus arcus AG, vel CG, ad GQ, ſinum rectum arcus GL. </s>
            <s xml:id="echoid-s13861" xml:space="preserve">Cum ergo ſinus BP,
              <lb/>
            GQ, ſint perpendiculares ex punctis B, G, in planum circuli
              <emph style="sc">AECf</emph>
            , demiſſæ, pa-
              <lb/>
            tet propoſitum. </s>
            <s xml:id="echoid-s13862" xml:space="preserve">Quòd ſi vnum punctorum acceptum ſit B, ex vna parte ſectionis A,
              <lb/>
            & </s>
            <s xml:id="echoid-s13863" xml:space="preserve">alterum punctum acceptum ſit D, ex altera parte ſectionis
              <emph style="sc">A</emph>
            , in eodem circulo;
              <lb/>
            </s>
            <s xml:id="echoid-s13864" xml:space="preserve">erit nibilominus ita ſinus arcus AB, ad perpendicularem BP, ex B, demiſſam in pla-
              <lb/>
            num alterius circuli AECF, vt ſinus arcus AD, ad perpendicularem, quæ ex D, in
              <lb/>
            planum alterius circuli AECF, demitteretur: </s>
            <s xml:id="echoid-s13865" xml:space="preserve">propterea quod oſtenſum eſt, ita eſſe
              <lb/>
            ſinum arcus AB, ad ſinum arcus BI, vt @ſt ſinus arcus AD, ad ſinum arcus DF; </s>
            <s xml:id="echoid-s13866" xml:space="preserve">qui
              <lb/>
            quidem ſinus arcuum BI, DF, ſunt perpendiculares ex punctis B, D, in planum
              <lb/>
            circuli AECF, cadentes, vt ex demonſtratis in hac propoſ. </s>
            <s xml:id="echoid-s13867" xml:space="preserve">liquido conſtat. </s>
            <s xml:id="echoid-s13868" xml:space="preserve">Idem
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            perſpicitur in figura poſteriori; </s>
            <s xml:id="echoid-s13869" xml:space="preserve">cum ibi etiam ſit, vt ſinus arcus AB, ad </s>
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