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

Table of handwritten notes

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            <s xml:id="echoid-s15076" xml:space="preserve">
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            menti anguli quæſiti. </s>
            <s xml:id="echoid-s15077" xml:space="preserve">Hincex arcu AB, & </s>
            <s xml:id="echoid-s15078" xml:space="preserve">vtroque angulo B, A, vter-
              <lb/>
              <figure xlink:label="fig-442-01" xlink:href="fig-442-01a" number="297">
                <image file="442-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/442-01"/>
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            que arcus AC, CB, inuenietur, vt in 2. </s>
            <s xml:id="echoid-s15079" xml:space="preserve">proble-
              <lb/>
            mate propoſ. </s>
            <s xml:id="echoid-s15080" xml:space="preserve">41. </s>
            <s xml:id="echoid-s15081" xml:space="preserve">oſtendimus.</s>
            <s xml:id="echoid-s15082" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15083" xml:space="preserve">AN vero angulus quæſitus
              <emph style="sc">A</emph>
            , acutus ſit, obtuſusve,
              <lb/>
            diſcemus exarcu dato
              <emph style="sc">AB</emph>
            , & </s>
            <s xml:id="echoid-s15084" xml:space="preserve">dato angulo
              <emph style="sc">B</emph>
            . </s>
            <s xml:id="echoid-s15085" xml:space="preserve">Nam ſi
              <emph style="sc">AB</emph>
            ,
              <lb/>
            eſt quadrante minor, & </s>
            <s xml:id="echoid-s15086" xml:space="preserve">angulus
              <emph style="sc">B</emph>
            , acutus quidem, erit
              <lb/>
              <note position="left" xlink:label="note-442-01" xlink:href="note-442-01a" xml:space="preserve">38. huius</note>
            & </s>
            <s xml:id="echoid-s15087" xml:space="preserve">
              <emph style="sc">A</emph>
            , acutus; </s>
            <s xml:id="echoid-s15088" xml:space="preserve">ſi autem
              <emph style="sc">B</emph>
            , eſt obtuſus, erit & </s>
            <s xml:id="echoid-s15089" xml:space="preserve">
              <emph style="sc">A</emph>
            , obtuſus.
              <lb/>
            </s>
            <s xml:id="echoid-s15090" xml:space="preserve">At ſi
              <emph style="sc">AB</emph>
            , eſt maior quadrante, & </s>
            <s xml:id="echoid-s15091" xml:space="preserve">
              <emph style="sc">B</emph>
            , quidem acutus, erit
              <lb/>
              <emph style="sc">A</emph>
            , obtuſus; </s>
            <s xml:id="echoid-s15092" xml:space="preserve">ſi vero
              <emph style="sc">B</emph>
            , eſt obtuſus, erit
              <emph style="sc">A</emph>
            , acutus.</s>
            <s xml:id="echoid-s15093" xml:space="preserve"/>
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        <div xml:id="echoid-div1213" type="section" level="1" n="575">
          <head xml:id="echoid-head610" xml:space="preserve">THEOR. 46. PROPOS. 48.</head>
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            <s xml:id="echoid-s15094" xml:space="preserve">IN omni triangulo ſphærico rectangulo, cuius
              <lb/>
            omnes arcus quadrante ſint minores: </s>
            <s xml:id="echoid-s15095" xml:space="preserve">Sinus totus
              <lb/>
            ad ſinum vtriusvis arcuum circa angulum rectum
              <lb/>
            eandem habet proportionem, quam tangens com
              <lb/>
            plementi alterius arcus circa angulum rectum ad
              <lb/>
            tangentem complementi anguli oppoſiti.</s>
            <s xml:id="echoid-s15096" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15097" xml:space="preserve">IN triangulo ſphærico ABC, cuius omnes arcus minores quadrante, ſit
              <lb/>
            rectus angulus B. </s>
            <s xml:id="echoid-s15098" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus AB, vt eſt tangens
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              <figure xlink:label="fig-442-02" xlink:href="fig-442-02a" number="298">
                <image file="442-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/442-02"/>
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            complementi arcus BC, ad tangentem com-
              <lb/>
            plementi anguli A. </s>
            <s xml:id="echoid-s15099" xml:space="preserve">Facta enim conſtructio-
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            ne, vt in propoſ. </s>
            <s xml:id="echoid-s15100" xml:space="preserve">45. </s>
            <s xml:id="echoid-s15101" xml:space="preserve">erit angulus D, rectus,
              <lb/>
            & </s>
            <s xml:id="echoid-s15102" xml:space="preserve">
              <emph style="sc">Cf</emph>
            , complementum arcus BC; </s>
            <s xml:id="echoid-s15103" xml:space="preserve">& </s>
            <s xml:id="echoid-s15104" xml:space="preserve">EF, com
              <lb/>
            plementum anguli A; </s>
            <s xml:id="echoid-s15105" xml:space="preserve">& </s>
            <s xml:id="echoid-s15106" xml:space="preserve">AD, quadrans, vt ibi
              <lb/>
            oſtenſum eſt. </s>
            <s xml:id="echoid-s15107" xml:space="preserve">Quoniam igitur duo circuli ma-
              <lb/>
            ximi AD, AE, in ſphæra ſe mutuo ſecãt in A,
              <lb/>
            ductiq́; </s>
            <s xml:id="echoid-s15108" xml:space="preserve">ſunt ex punctis C, E, ad arcum AD,
              <lb/>
            arcus perpendiculares CB, ED; </s>
            <s xml:id="echoid-s15109" xml:space="preserve">erit, vt ſinus
              <lb/>
            totus quadrantis AD, ad tangentem arcus
              <lb/>
              <note position="left" xlink:label="note-442-02" xlink:href="note-442-02a" xml:space="preserve">Theor. 6.
                <lb/>
              ſcholij 40.
                <lb/>
              huius.</note>
            DE, ita ſinus arcus AB, ad tangentem arcus
              <lb/>
            BC: </s>
            <s xml:id="echoid-s15110" xml:space="preserve">Et permutando, vt ſinus totus ad ſinum
              <lb/>
            arcus AB, ita tangens arcus DE, ad tangen-
              <lb/>
            tem arcus BC. </s>
            <s xml:id="echoid-s15111" xml:space="preserve">Eſt autem, (cum CF, EF, ſint complementa arcuum BC, DE,)
              <lb/>
            vt tangens arcus DE, ad tangentem arcus BC, ita tangens arcus CF, ad tan
              <lb/>
              <note position="left" xlink:label="note-442-03" xlink:href="note-442-03a" xml:space="preserve">81. Sinuũ</note>
            gentem arcus EF. </s>
            <s xml:id="echoid-s15112" xml:space="preserve">Igitur erit quoque, vt ſinus totus ad ſinum arcus AB, ita
              <lb/>
            tangens arcus CF, hoc eſt, complementi arcus BC, ad tangentem arcus EF,
              <lb/>
            hoceſt, complementi anguli A, arcui BC, oppoſiti. </s>
            <s xml:id="echoid-s15113" xml:space="preserve">Non aliter oſtendemus,
              <lb/>
            ſi aliter figura conſtruatur, ita eſſe ſinum totum ad ſinum arcus BC, vt eſt tan
              <lb/>
            gens complementi arcus AB, ad tangentem complementi anguli C. </s>
            <s xml:id="echoid-s15114" xml:space="preserve">In omni
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            triangulo ergo ſphærico rectangulo, &</s>
            <s xml:id="echoid-s15115" xml:space="preserve">c. </s>
            <s xml:id="echoid-s15116" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s15117" xml:space="preserve"/>
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