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

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        <div xml:id="echoid-div1170" type="section" level="1" n="561">
          <p style="it">
            <s xml:id="echoid-s14744" xml:space="preserve">
              <pb o="422" file="434" n="434" rhead=""/>
            gemus ſequentes propoſitiones ad triangula quoque ſphærica rectangula ſpectantes,
              <lb/>
            antequam triangulorum ſphæricorum non rectangulorum calculum exponamus.
              <lb/>
            </s>
            <s xml:id="echoid-s14745" xml:space="preserve">Vt autem clariores fiant demonſtrationes, & </s>
            <s xml:id="echoid-s14746" xml:space="preserve">minus confuſæ, proponemus ſemper
              <lb/>
            triangulum ſphæricum rectangulum, cuius duo arcus circa angulum rectum, ac pro-
              <lb/>
            inde omnes tres, minores ſint quadrante. </s>
            <s xml:id="echoid-s14747" xml:space="preserve">Nam eædem demonſtrationes alijs omnibus
              <lb/>
            conuenient, vt in hoc ſcbolio demonſtrauimus: </s>
            <s xml:id="echoid-s14748" xml:space="preserve">quippe cum & </s>
            <s xml:id="echoid-s14749" xml:space="preserve">tam duo arcus ſemicir
              <lb/>
            culum conſicientes, quàm duo anguli duobus rectis æquales, eandem habeant tangen
              <lb/>
            tem, ac ſecantem, quemadmodum & </s>
            <s xml:id="echoid-s14750" xml:space="preserve">eundem ſinum, vt in tractatione tangentium,
              <lb/>
            & </s>
            <s xml:id="echoid-s14751" xml:space="preserve">ſecantium monuimus.</s>
            <s xml:id="echoid-s14752" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1173" type="section" level="1" n="562">
          <head xml:id="echoid-head597" xml:space="preserve">THEOR. 42. PROPOS. 44.</head>
          <p>
            <s xml:id="echoid-s14753" xml:space="preserve">IN omni triangulo ſphætico rectangulo, cu-
              <lb/>
            ius omnes arcus quadrante ſint minores: </s>
            <s xml:id="echoid-s14754" xml:space="preserve">ſinus to-
              <lb/>
            tus ad ſinum vtriuſuis arcuum circarectum angu-
              <lb/>
            lum eandem habet proportionem, quam tangens
              <lb/>
            anguli non recti dicto arcui adiacentis ad tangen-
              <lb/>
            tem reliqui arcus circa angulum rectum huic an-
              <lb/>
            gulo oppoſiti.</s>
            <s xml:id="echoid-s14755" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14756" xml:space="preserve">IN triangulo ſph ærico ABC, cuius omnes arcus quadrante minores, ſit
              <lb/>
            angulus C, rectus. </s>
            <s xml:id="echoid-s14757" xml:space="preserve">Dico ita eſſe ſinum totum ad ſinum arcus BC, vt eſt tan-
              <lb/>
            gens anguli B, ad tangentem arcus AC. </s>
            <s xml:id="echoid-s14758" xml:space="preserve">Productis
              <lb/>
              <figure xlink:label="fig-434-01" xlink:href="fig-434-01a" number="288">
                <image file="434-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/434-01"/>
              </figure>
            enim arcubus BC, BA, donec fiant quadrantes BF,
              <lb/>
            BD, ac per puncta F, D, arcu FD, circuli maximi
              <lb/>
            deſcripto; </s>
            <s xml:id="echoid-s14759" xml:space="preserve">erit vterque angulus F, D, rectus, ob qua-
              <lb/>
              <note position="left" xlink:label="note-434-01" xlink:href="note-434-01a" xml:space="preserve">25. huius.</note>
            drantes BF, BD: </s>
            <s xml:id="echoid-s14760" xml:space="preserve">& </s>
            <s xml:id="echoid-s14761" xml:space="preserve">DF, arcus erit anguli B; </s>
            <s xml:id="echoid-s14762" xml:space="preserve">cum
              <lb/>
            B, polus ſit arcus DF. </s>
            <s xml:id="echoid-s14763" xml:space="preserve">Quia igitur duo circuli ma-
              <lb/>
              <note position="left" xlink:label="note-434-02" xlink:href="note-434-02a" xml:space="preserve">26. huius.</note>
            ximi in ſphæra BF, BD, ſecant ſeſe in B, ductiq́ue
              <lb/>
            ſunt ex A, D, ad BF, arcus perpendiculares AC,
              <lb/>
            DF; </s>
            <s xml:id="echoid-s14764" xml:space="preserve">erit, vt ſinus quadrantis BF, hoc eſt, ſinus to-
              <lb/>
              <note position="left" xlink:label="note-434-03" xlink:href="note-434-03a" xml:space="preserve">Theor. 6.
                <lb/>
              ſcholij. 40.
                <lb/>
              huius.</note>
            tus, ad tangentem arcus FD, hoc eſt, ad tangentem
              <lb/>
            anguli B, ita ſinus arcus BC, ad tangentem arcus AC: </s>
            <s xml:id="echoid-s14765" xml:space="preserve">Et permutando, vt
              <lb/>
            ſinus totus ad ſinum arcus BC, ita tangens anguli B, ad tangentem arcus AC.
              <lb/>
            </s>
            <s xml:id="echoid-s14766" xml:space="preserve">Non aliter demonſtrabimus, ita eſſe ſinum totum ad ſinum arcus AC, vt eſt
              <lb/>
            tangens anguli A, ad tangentem arcus BC: </s>
            <s xml:id="echoid-s14767" xml:space="preserve">vt patet, ſi arcus AC, AB, pro-
              <lb/>
            ducantur, donec fiant quadrantes AG, AE, perque G, E, arcus maximi cir-
              <lb/>
            culi deſcribatur GE. </s>
            <s xml:id="echoid-s14768" xml:space="preserve">Erit enim rurſus, vt ſinus quadrantis AG, id eſt, ſinus
              <lb/>
              <note position="left" xlink:label="note-434-04" xlink:href="note-434-04a" xml:space="preserve">Theor. 6.
                <lb/>
              ſcholij 40.
                <lb/>
              huius.</note>
            totus, ad tangentem arcus EG, ſeu anguli A, ita ſinus arcus AC, ad tangen
              <lb/>
            tem arcus BC: </s>
            <s xml:id="echoid-s14769" xml:space="preserve">Et permutãdo, vt ſinus totus ad ſinum arcus AC, ita tangens
              <lb/>
            anguli A, ad tangentem arcus BC. </s>
            <s xml:id="echoid-s14770" xml:space="preserve">In omni ergo triangulo ſphærico rectan-
              <lb/>
            gulo, &</s>
            <s xml:id="echoid-s14771" xml:space="preserve">c. </s>
            <s xml:id="echoid-s14772" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s14773" xml:space="preserve"/>
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