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

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          <p>
            <s xml:id="echoid-s14225" xml:space="preserve">
              <pb o="411" file="423" n="423" rhead=""/>
            gulus D, rectus ſit; </s>
            <s xml:id="echoid-s14226" xml:space="preserve">eſt, vt ſinus arcus AD, ad ſinum anguli ACD, ita ſinus
              <lb/>
            arcus AC, ad ſinum anguli ADC: </s>
            <s xml:id="echoid-s14227" xml:space="preserve">& </s>
            <s xml:id="echoid-s14228" xml:space="preserve">permutando, vt ſinus arcus AD, ad ſi-
              <lb/>
            num arcus AC, ita ſinus anguli ACD, ad ſinum anguli ADC, hoc eſt, ad
              <lb/>
            ſinum anguli ADB, cum anguli ad D, ſint recti. </s>
            <s xml:id="echoid-s14229" xml:space="preserve">Ex æqualitate ergo, & </s>
            <s xml:id="echoid-s14230" xml:space="preserve">per-
              <lb/>
            turbata proportione, erit, vt ſinus arcus AB, ad ſinum arcus AC, ita ſinus an-
              <lb/>
            guli ACD, ad ſinum anguli B; </s>
            <s xml:id="echoid-s14231" xml:space="preserve">vt in appoſita formula apparet. </s>
            <s xml:id="echoid-s14232" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s14233" xml:space="preserve">per-
              <lb/>
            mutando erit, vt ſinus arcus AB, in triangulo ABC, ad ſinum anguli ACB,
              <lb/>
            in eodem triangulo ABC, ita ſinus arcus AC, eiuſdem
              <lb/>
            trianguli ABC, ad ſinum anguli B, in eodem triangulo ABC.</s>
            <s xml:id="echoid-s14234" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">
            <lb/>
          arcus # anguli
            <lb/>
          AB. # ACD.
            <lb/>
          AD. # ADB.
            <lb/>
          AC. # B.
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s14235" xml:space="preserve">CADAT deinde arcus per A, & </s>
            <s xml:id="echoid-s14236" xml:space="preserve">polum circuli BC,
              <lb/>
            ductus in arcum BC, productum ad E, eritq́; </s>
            <s xml:id="echoid-s14237" xml:space="preserve">angulus E, re-
              <lb/>
              <note position="right" xlink:label="note-423-02" xlink:href="note-423-02a" xml:space="preserve">15. 1. Theod.</note>
            ctus. </s>
            <s xml:id="echoid-s14238" xml:space="preserve">Quoniam igitur in triangulo ABE, angulus E, rectus
              <lb/>
            eſt; </s>
            <s xml:id="echoid-s14239" xml:space="preserve">erit, vt demonſtratum eſt, vt ſinus arcus AB, ad ſinum
              <lb/>
            anguli E, ita ſinus arcus AE, ad ſinum anguli B: </s>
            <s xml:id="echoid-s14240" xml:space="preserve">& </s>
            <s xml:id="echoid-s14241" xml:space="preserve">permu-
              <lb/>
            tando, vt ſinus arcus AB, ad ſinum arcus AE, ita ſinus anguli E, ad ſinum
              <lb/>
            anguli B. </s>
            <s xml:id="echoid-s14242" xml:space="preserve">Sed eadem ratione, cum in triangulo ACE, angulus E, rectus ſit,
              <lb/>
            eſt, vt ſinus arcus AE, ad ſinum anguli ACE, ita ſinus arcus AC, ad ſi-
              <lb/>
            num anguli E: </s>
            <s xml:id="echoid-s14243" xml:space="preserve">& </s>
            <s xml:id="echoid-s14244" xml:space="preserve">permutando, vt ſinus arcus AE, ad ſinum arcus AC, ita
              <lb/>
            ſinus anguli ACE, ad ſinum anguli E. </s>
            <s xml:id="echoid-s14245" xml:space="preserve">Igitur ex æqualitate, & </s>
            <s xml:id="echoid-s14246" xml:space="preserve">perturbata
              <lb/>
            proportione, erit vt ſinus arcus AB, ad ſinum arcus AC,
              <lb/>
            ita ſinus anguli ACE, hoc eſt, anguli ACB, (cum idem ſit
              <lb/>
              <note position="right" xlink:label="note-423-03" xlink:href="note-423-03a" xml:space="preserve">
                <lb/>
              arcus # anguli
                <lb/>
              AB. # ACE.
                <lb/>
              AE. # E.
                <lb/>
              AC. # B.
                <lb/>
              </note>
            ſinus vtriuſq; </s>
            <s xml:id="echoid-s14247" xml:space="preserve">anguli ad C, quòd eorum arcus ſemicirculum
              <lb/>
            conſtituant, vt conſtat ex coroll. </s>
            <s xml:id="echoid-s14248" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s14249" xml:space="preserve">5. </s>
            <s xml:id="echoid-s14250" xml:space="preserve">huius tracta-
              <lb/>
            tus. </s>
            <s xml:id="echoid-s14251" xml:space="preserve">Perſpicuum autem eſt ex ijs, quæ in tractatione ſinuum
              <lb/>
            diximus, duos arcus ſemicirculum conficientes, eundem ha-
              <lb/>
            bere ſinum.) </s>
            <s xml:id="echoid-s14252" xml:space="preserve">ad ſinum anguli B; </s>
            <s xml:id="echoid-s14253" xml:space="preserve">vt in appoſita ſormula ap-
              <lb/>
            paret. </s>
            <s xml:id="echoid-s14254" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s14255" xml:space="preserve">permutando erit, vt ſinus arcus AB, in triangulo ABC,
              <lb/>
            ad ſinum anguli ACB, eiuſdem trianguli ABC, ita ſinus arcus AC, in
              <lb/>
            eodem triangulo ABC, ad ſinum anguli B, eiuſdem trianguli ABC. </s>
            <s xml:id="echoid-s14256" xml:space="preserve">Quod
              <lb/>
            ſi ex B, ad arcum AC, ducatur alius arcus perpendicularis, qui vel intra trian
              <lb/>
            gulum cadet, vel in arcum productum, oſtendemus eodem modo, ita eſſe ſinũ
              <lb/>
            arcus AB, ad ſinum anguli ACB, vt eſt ſinus arcus BC, ad ſinum anguli
              <lb/>
            BAC. </s>
            <s xml:id="echoid-s14257" xml:space="preserve">Itaque in omni triangulo ſphærico, ſinus cuiuſlibet arcus, &</s>
            <s xml:id="echoid-s14258" xml:space="preserve">c. </s>
            <s xml:id="echoid-s14259" xml:space="preserve">Quod
              <lb/>
            erat oſtendendum.</s>
            <s xml:id="echoid-s14260" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1127" type="section" level="1" n="550">
          <head xml:id="echoid-head585" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s14261" xml:space="preserve">HINC perſpicuum eſt, in omni triangulo ſphærico rectangulo, vt eſt ſinus arcus rectum
              <lb/>
            angulum ſubtendentis ad ſinum totum, nempe ad ſinum anguli recti, quem ſubtendit, ita
              <lb/>
            eſſe ſinum cuiuſ@@bet reliquorum arcuum ad ſinum anguli, quem ſubtendit. </s>
            <s xml:id="echoid-s14262" xml:space="preserve">Quod idcir.
              <lb/>
            </s>
            <s xml:id="echoid-s14263" xml:space="preserve">co dixerim, quia plerique ſcriptores hoc corollarium, tanquàm propoſitionem ab hac no-
              <lb/>
            ftra propoſitione 41. </s>
            <s xml:id="echoid-s14264" xml:space="preserve">diuerſam, demonſtrant: </s>
            <s xml:id="echoid-s14265" xml:space="preserve">ſed placuit nobis propoſitionem hanc magis
              <lb/>
            vniuerſalem reddere, prout nimirum complectitur & </s>
            <s xml:id="echoid-s14266" xml:space="preserve">triangulum ſphæricum rectangulum,
              <lb/>
            & </s>
            <s xml:id="echoid-s14267" xml:space="preserve">non rectangulum.</s>
            <s xml:id="echoid-s14268" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1128" type="section" level="1" n="551">
          <head xml:id="echoid-head586" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s14269" xml:space="preserve">IN hac, & </s>
            <s xml:id="echoid-s14270" xml:space="preserve">ſequentibus propoſitionibus adducemus problemata, quibus ſphæri-
              <lb/>
            corum triangulorum rectangulorum calculus perficitur, quæq́; </s>
            <s xml:id="echoid-s14271" xml:space="preserve">ex ipſis propoſitioni-
              <lb/>
            bus eliciuntur, Quanquam autem nonnunquam in problemate aliquo plura </s>
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