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

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          <p>
            <s xml:id="echoid-s14186" xml:space="preserve">
              <pb o="410" file="422" n="422" rhead=""/>
            vterque angulus E, G, rectus erit. </s>
            <s xml:id="echoid-s14187" xml:space="preserve">Quia igitur duo maximi circuli BD, BC,
              <lb/>
            ſe mutuo in ſphæra ſecant in B, ſumptaq́ue ſunt in BD, duo puncta A, D, à
              <lb/>
            quibus ad BC, ducti ſunt duo arcus AC, DF, perpendiculares, erit vt ſinus
              <lb/>
            arcus AB, ad ſinum arcus AC, ita ſinus arcus BD, ad ſin um arcus DF: </s>
            <s xml:id="echoid-s14188" xml:space="preserve">& </s>
            <s xml:id="echoid-s14189" xml:space="preserve">per-
              <lb/>
              <note position="left" xlink:label="note-422-01" xlink:href="note-422-01a" xml:space="preserve">40. huius.</note>
            mutando, vt ſinus arcus AB, trianguli ABC, ad ſinum quadrantis BD, hoc
              <lb/>
            eſt, ad ſinum totum anguli recti C, in eodem triangulo ABC, ita ſinus arcus
              <lb/>
            AC, eiuſdem trianguli ABC, ad ſinum arcus DF, hoc eſt, ad ſinum anguli
              <lb/>
            B, in eodem triangulo ABC. </s>
            <s xml:id="echoid-s14190" xml:space="preserve">Eademq; </s>
            <s xml:id="echoid-s14191" xml:space="preserve">ratione erit, vt ſinus arcus AB, trian
              <lb/>
            guli ABC, ad ſinum quadrantis AE, hoc eſt, ad ſinum totum anguli recti C,
              <lb/>
            in eodem triangulo ABC, ita finus arcus BC, eiuſdem trianguli ABC, ad
              <lb/>
            finum arcus EG, hoc eſt, ad ſinum anguli A, in eodem triangulo ABC. </s>
            <s xml:id="echoid-s14192" xml:space="preserve">Quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s14193" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14194" xml:space="preserve">SI denique alter angulorum A, B, recto maior eſt, & </s>
            <s xml:id="echoid-s14195" xml:space="preserve">alter minor; </s>
            <s xml:id="echoid-s14196" xml:space="preserve">ſit B, ma-
              <lb/>
            ior, & </s>
            <s xml:id="echoid-s14197" xml:space="preserve">A, minor. </s>
            <s xml:id="echoid-s14198" xml:space="preserve">Erit igitur arcus AB, quadrante maior: </s>
            <s xml:id="echoid-s14199" xml:space="preserve">Item arcus AC,
              <lb/>
              <note position="left" xlink:label="note-422-02" xlink:href="note-422-02a" xml:space="preserve">37. huius.</note>
              <note position="left" xlink:label="note-422-03" xlink:href="note-422-03a" xml:space="preserve">34. huius.</note>
            quadrante etiam maior, at verò BC, minor
              <lb/>
            quadrante. </s>
            <s xml:id="echoid-s14200" xml:space="preserve">Abſcindantur ergo quadrantes
              <lb/>
              <figure xlink:label="fig-422-01" xlink:href="fig-422-01a" number="273">
                <image file="422-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/422-01"/>
              </figure>
            BD, AE, & </s>
            <s xml:id="echoid-s14201" xml:space="preserve">AG, productoq́ue arcu BC, fiat
              <lb/>
              <note position="left" xlink:label="note-422-04" xlink:href="note-422-04a" xml:space="preserve">20. 1 Theod.</note>
            quadrans BF; </s>
            <s xml:id="echoid-s14202" xml:space="preserve">& </s>
            <s xml:id="echoid-s14203" xml:space="preserve">per puncta D, F, ducatur ar-
              <lb/>
            cus DF, circuli maximi, necnon per E, G,
              <lb/>
            arcus circuli maximi EG; </s>
            <s xml:id="echoid-s14204" xml:space="preserve">eritq́ue rurſus B,
              <lb/>
            polus arcus DF, & </s>
            <s xml:id="echoid-s14205" xml:space="preserve">A, polus arcus EG. </s>
            <s xml:id="echoid-s14206" xml:space="preserve">Igi-
              <lb/>
              <note position="left" xlink:label="note-422-05" xlink:href="note-422-05a" xml:space="preserve">26. huius.</note>
            tur DF, EG, arcus erunt angulorum B, A;
              <lb/>
            </s>
            <s xml:id="echoid-s14207" xml:space="preserve">necnon tam quadrans BD, quam AE, arcus
              <lb/>
            anguli C, recti, ex defin. </s>
            <s xml:id="echoid-s14208" xml:space="preserve">6. </s>
            <s xml:id="echoid-s14209" xml:space="preserve">Item propter qua-
              <lb/>
            drantes AE, AG, vterque angulus E, G, re-
              <lb/>
              <note position="left" xlink:label="note-422-06" xlink:href="note-422-06a" xml:space="preserve">25. huius.</note>
            ctus erit. </s>
            <s xml:id="echoid-s14210" xml:space="preserve">Quoniam igitur duo circuli maxi-
              <lb/>
            mi BA, BF, in ſphæra ſe mutuo ſecant in B,
              <lb/>
            ſumptaq́ue ſunt in BA, duo puncta A, D, à
              <lb/>
            quibus ad BF, ducti ſunt duo arcus perpendiculares AC, DF; </s>
            <s xml:id="echoid-s14211" xml:space="preserve">erit, vt ſinus
              <lb/>
            arcus AB, ad ſinum arcus AC, ita ſinus arcus BD, ad ſinum arcus DF: </s>
            <s xml:id="echoid-s14212" xml:space="preserve">& </s>
            <s xml:id="echoid-s14213" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-422-07" xlink:href="note-422-07a" xml:space="preserve">40. huius.</note>
            permutando, vt ſinus arcus AB, trianguli ABC, ad ſinum quadrantis BD,
              <lb/>
            hoc eſt, ad ſinum totum anguli recti C, in eodem triangulo ABC, ita ſinus
              <lb/>
            arcus AC, trianguli eiuſdem ABC, ad ſinum arcus DF, hoc eſt, ad ſinum an-
              <lb/>
            guli B, in triangulo eodem ABC. </s>
            <s xml:id="echoid-s14214" xml:space="preserve">Eodemq́ue modo erit, vt ſinus arcus AB,
              <lb/>
            trianguli ABC, ad ſinum quadrantis AE, hoc eſt, ad ſinum totum recti an-
              <lb/>
            guli C, in eodẽ triangulo ABC, ita ſinus arcus BC, eiuſdem trianguli ABC,
              <lb/>
            ad ſinum arcus EG, hoc eſt, ad ſinum anguli A, eiuſdem trianguli ABC,
              <lb/>
            Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14215" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s14216" xml:space="preserve">QVARTO ac vltimo nullus angulorum A, B, C, rectus ſit. </s>
            <s xml:id="echoid-s14217" xml:space="preserve">Per pun-
              <lb/>
              <figure xlink:label="fig-422-02" xlink:href="fig-422-02a" number="274">
                <image file="422-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/422-02"/>
              </figure>
            ctum A, & </s>
            <s xml:id="echoid-s14218" xml:space="preserve">polum circuli BC, ducatur arcus circu-
              <lb/>
              <note position="left" xlink:label="note-422-08" xlink:href="note-422-08a" xml:space="preserve">20. 1 Theod.</note>
            li maximi AD, cadatq́ue primum in latus BC, in-
              <lb/>
              <note position="left" xlink:label="note-422-09" xlink:href="note-422-09a" xml:space="preserve">25. 1. Theod.</note>
            tra triangulum; </s>
            <s xml:id="echoid-s14219" xml:space="preserve">eruntq́; </s>
            <s xml:id="echoid-s14220" xml:space="preserve">anguli ad D, recti. </s>
            <s xml:id="echoid-s14221" xml:space="preserve">Quoniam
              <lb/>
            igitur in triangulo ABD, angulus D, rectus eſt; </s>
            <s xml:id="echoid-s14222" xml:space="preserve">erit,
              <lb/>
            vt iam demonſtratum eſt, vt ſinus arcus AB, ad ſi-
              <lb/>
            num anguli ADB, ita ſinus arcus AD, ad ſinum an-
              <lb/>
            guli B: </s>
            <s xml:id="echoid-s14223" xml:space="preserve">& </s>
            <s xml:id="echoid-s14224" xml:space="preserve">permutando, vt ſinus arcus AB, ad ſinum
              <lb/>
            arcus AD, ita ſinus anguli ADB, ad ſinum anguli
              <lb/>
            B. </s>
            <s xml:id="echoid-s14225" xml:space="preserve">Sed eodem modo, cum in triangulo ADC, </s>
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