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

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        <div xml:id="echoid-div1115" type="section" level="1" n="548">
          <p style="it">
            <s xml:id="echoid-s14097" xml:space="preserve">
              <pb o="408" file="420" n="420" rhead=""/>
            tem arcus
              <emph style="sc">EF</emph>
            . </s>
            <s xml:id="echoid-s14098" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s14099" xml:space="preserve">proportio ſecantis arcus CF, ad ſecantem arcus
              <emph style="sc">E</emph>
            F,
              <lb/>
            componatur ex proportione ſecantis arcus CF, ad ſinum totum, & </s>
            <s xml:id="echoid-s14100" xml:space="preserve">ex proportione
              <lb/>
            ſinus totius ad ſecantem arcus
              <emph style="sc">EF</emph>
            ; </s>
            <s xml:id="echoid-s14101" xml:space="preserve">quòd ſinus totus ſit medius inter has ſecãtes: </s>
            <s xml:id="echoid-s14102" xml:space="preserve">erit,
              <lb/>
            vt ſinus arcus
              <emph style="sc">AE</emph>
            , ad ſinum arcus
              <emph style="sc">AC</emph>
            , ita ſecans arcus CF, ad ſecantem arcus
              <emph style="sc">EF</emph>
            ;
              <lb/>
            </s>
            <s xml:id="echoid-s14103" xml:space="preserve">& </s>
            <s xml:id="echoid-s14104" xml:space="preserve">permutando, vt ſinus arcus
              <emph style="sc">AE</emph>
            , ad ſecantem arcus CF, ita ſinus arcus AC, ad
              <lb/>
            ſecantem arcus
              <emph style="sc">EF</emph>
            . </s>
            <s xml:id="echoid-s14105" xml:space="preserve">Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14106" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14107" xml:space="preserve">ALITER. </s>
            <s xml:id="echoid-s14108" xml:space="preserve">Quoniam eſt, vt ſinus arcus
              <emph style="sc">AE</emph>
            , ad ſinum arcus
              <emph style="sc">E</emph>
            D, ita ſinus ar-
              <lb/>
              <note position="left" xlink:label="note-420-01" xlink:href="note-420-01a" xml:space="preserve">40. huius.</note>
            cus
              <emph style="sc">AC</emph>
            , ad ſinum arcus
              <emph style="sc">CB</emph>
            ; </s>
            <s xml:id="echoid-s14109" xml:space="preserve">hoc eſt, permutando, vt ſinus arcus
              <emph style="sc">AE</emph>
            , ad ſinum ar-
              <lb/>
            cus
              <emph style="sc">AC</emph>
            , ita ſinus arcus
              <emph style="sc">E</emph>
            D, ad ſinum arcus
              <emph style="sc">CB</emph>
            : </s>
            <s xml:id="echoid-s14110" xml:space="preserve">
              <emph style="sc">E</emph>
            ſt autemita ſecans arcus
              <emph style="sc">CF</emph>
            ,
              <lb/>
              <note position="left" xlink:label="note-420-02" xlink:href="note-420-02a" xml:space="preserve">22. Sinuũ.</note>
            ad ſecantem arcus
              <emph style="sc">E</emph>
            F, vt ſinus
              <emph style="sc">E</emph>
            D, qui complementum eſt poſterioris arcus
              <emph style="sc">EF</emph>
            , ad
              <lb/>
            ſinum arcus
              <emph style="sc">CB</emph>
            , qui complementum eſt arcus prioris
              <emph style="sc">CF</emph>
            ; </s>
            <s xml:id="echoid-s14111" xml:space="preserve">erit quoque, vt ſinus ar-
              <lb/>
            cus
              <emph style="sc">AE</emph>
            , ad ſinum arcus
              <emph style="sc">AC</emph>
            , ita ſecans arcus CF, ad ſecantem arcus
              <emph style="sc">EF</emph>
            . </s>
            <s xml:id="echoid-s14112" xml:space="preserve">
              <emph style="sc">E</emph>
            t per-
              <lb/>
            mutando, vt ſinus arcus
              <emph style="sc">AE</emph>
            , ad ſecantem arcus CF, ita ſinus arcus
              <emph style="sc">AC</emph>
            , ad ſecan-
              <lb/>
            tem arcus
              <emph style="sc">EF</emph>
            . </s>
            <s xml:id="echoid-s14113" xml:space="preserve">Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14114" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14115" xml:space="preserve">EADEM hæc demonſtratio locum etiam habet, licet duo puncta aſſumpta ſint ad
              <lb/>
            diuerſas partes puncti ſectionis. </s>
            <s xml:id="echoid-s14116" xml:space="preserve">Secent enim rurſum ſeſe duo circuli maximi
              <emph style="sc">EF, BA</emph>
            ,
              <lb/>
            in D; </s>
            <s xml:id="echoid-s14117" xml:space="preserve">& </s>
            <s xml:id="echoid-s14118" xml:space="preserve">à punctis
              <emph style="sc">F, E</emph>
            , arcus
              <emph style="sc">EF</emph>
            , ducantur ad
              <emph style="sc">BA</emph>
            , arcus perpendiculares
              <emph style="sc">FA, EB</emph>
            .
              <lb/>
            </s>
            <s xml:id="echoid-s14119" xml:space="preserve">Dicoita eſſe ſinum arcus
              <emph style="sc">E</emph>
            D, ad ſecantem complementi
              <lb/>
            arcus
              <emph style="sc">FA</emph>
            , vt est ſinus arcus DF, ad ſecantem comple-
              <lb/>
              <figure xlink:label="fig-420-01" xlink:href="fig-420-01a" number="269">
                <image file="420-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/420-01"/>
              </figure>
            menti arcus
              <emph style="sc">EB</emph>
            . </s>
            <s xml:id="echoid-s14120" xml:space="preserve">Nam quoniam eſt, vt ſinus arcus ED,
              <lb/>
              <note position="left" xlink:label="note-420-03" xlink:href="note-420-03a" xml:space="preserve">40. huius.</note>
            ad ſinum arcus
              <emph style="sc">EB</emph>
            , ita ſinus arcus DF, ad ſinum arcus
              <lb/>
              <emph style="sc">FA</emph>
            ; </s>
            <s xml:id="echoid-s14121" xml:space="preserve">& </s>
            <s xml:id="echoid-s14122" xml:space="preserve">permutando, vt ſinus arcus
              <emph style="sc">E</emph>
            D, ad ſinum arcus
              <lb/>
            DF, ita ſinus arcus
              <emph style="sc">EB</emph>
            , ad ſinum arcus
              <emph style="sc">FA</emph>
            ; </s>
            <s xml:id="echoid-s14123" xml:space="preserve">Vt autem
              <lb/>
            ſinus arcus
              <emph style="sc">EB</emph>
            , ad ſinum arcus
              <emph style="sc">FA</emph>
            , ita eſt ſecans com-
              <lb/>
              <note position="left" xlink:label="note-420-04" xlink:href="note-420-04a" xml:space="preserve">22. Sinuũ.</note>
            plementi arcus
              <emph style="sc">FA</emph>
            , ad ſecantem complementi arcus
              <emph style="sc">EB</emph>
            :
              <lb/>
            </s>
            <s xml:id="echoid-s14124" xml:space="preserve">erit quoque, vt ſinus arcus
              <emph style="sc">E</emph>
            D, ad ſinum arcus DF, ita
              <lb/>
            ſecans complementi arcus
              <emph style="sc">FA</emph>
            , ad ſecantem complementi
              <lb/>
            arcus
              <emph style="sc">EB</emph>
            ; </s>
            <s xml:id="echoid-s14125" xml:space="preserve">& </s>
            <s xml:id="echoid-s14126" xml:space="preserve">permutando, vt ſinus arcus
              <emph style="sc">E</emph>
            D, ad ſecantem complementi arcus
              <emph style="sc">FA</emph>
            ,
              <lb/>
            ita ſinus arcus DF, ad ſecantem complementi arcus
              <emph style="sc">EB</emph>
            . </s>
            <s xml:id="echoid-s14127" xml:space="preserve">Quod eſt propoſi@um.</s>
            <s xml:id="echoid-s14128" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s14129" xml:space="preserve">REPETIVIMVS autem hic figuram quartam propoſ. </s>
            <s xml:id="echoid-s14130" xml:space="preserve">35. </s>
            <s xml:id="echoid-s14131" xml:space="preserve">licet arcuum BC,
              <lb/>
            CF, nulla fiat mentio, ne nouam figuram cogeremur extruere.</s>
            <s xml:id="echoid-s14132" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1119" type="section" level="1" n="549">
          <head xml:id="echoid-head584" xml:space="preserve">THEOR. 39. PROPOS. 41.</head>
          <p>
            <s xml:id="echoid-s14133" xml:space="preserve">IN omni triangulo ſphærico, ſinus cuiuſlibet
              <lb/>
            arcus ad ſinum anguli, quem ſubtendit, eandem
              <lb/>
            habet proportionem, quam ſinus vtriuſque reli-
              <lb/>
            quorum arcuum ad ſinũ anguli, quem ſubtendit.</s>
            <s xml:id="echoid-s14134" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14135" xml:space="preserve">SIT triangulum ſphæricum quodcunque ABC. </s>
            <s xml:id="echoid-s14136" xml:space="preserve">Dico ita eſſe ſinum ar-
              <lb/>
            cus AB, ad ſinum anguli C, quem ſubtendit, vt eſt ſinus arcus AC, ad ſinum
              <lb/>
            auguli B, quem ſubtendit, & </s>
            <s xml:id="echoid-s14137" xml:space="preserve">vt ſinus arcus BC, ad ſinum anguli A, quem
              <lb/>
            ſubtendit. </s>
            <s xml:id="echoid-s14138" xml:space="preserve">Sint enim primum omnes tres anguli recti; </s>
            <s xml:id="echoid-s14139" xml:space="preserve">eruntq́ue propterea
              <lb/>
              <note position="left" xlink:label="note-420-05" xlink:href="note-420-05a" xml:space="preserve">Coroll. 25.
                <lb/>
              huius.</note>
            omnes arcus quadrantes. </s>
            <s xml:id="echoid-s14140" xml:space="preserve">Manifeſtum igitur eſt, vt eſt ſinus totus </s>
          </p>
        </div>
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    </echo>
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