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

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        <div xml:id="echoid-div802" type="section" level="1" n="456">
          <pb o="310" file="322" n="322" rhead=""/>
          <p>
            <s xml:id="echoid-s9528" xml:space="preserve">IAM vero ſi detur duorum laterũ quorumlibet proportio, & </s>
            <s xml:id="echoid-s9529" xml:space="preserve">vnum latus,
              <lb/>
              <note position="left" xlink:label="note-322-01" xlink:href="note-322-01a" xml:space="preserve">Quãdo {pro}-
                <lb/>
              portio duo
                <lb/>
              rum laterũ
                <lb/>
              datur, & v-
                <lb/>
              nũlatus.</note>
            quodcũque illud ſit, ſumemus numeros proportionis notæ, ac ſi eſſent partes
              <lb/>
            alicuius menſurę, in quibus duo illa latera dentur; </s>
            <s xml:id="echoid-s9530" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s9531" xml:space="preserve">ex his, vt demonſtra-
              <lb/>
            uimus in hac propoſ. </s>
            <s xml:id="echoid-s9532" xml:space="preserve">angulos inueniemus, ac tertium latus in eiſdẽ partibus.
              <lb/>
            </s>
            <s xml:id="echoid-s9533" xml:space="preserve">Deinde, ſi ſiat, vt numerus illius lateris, quod datum eſt, ad ipſum latus datũ,
              <lb/>
            ita numeri aliorum laterum ſigillatim ad aliud, reperientur alia latera in par-
              <lb/>
            tibus menſuræ, ſecundum quam illud alterum latus eſt datum. </s>
            <s xml:id="echoid-s9534" xml:space="preserve">Vt ſi propor-
              <lb/>
            tio AB, ad AC, ſit, vt 15. </s>
            <s xml:id="echoid-s9535" xml:space="preserve">ad 39. </s>
            <s xml:id="echoid-s9536" xml:space="preserve">& </s>
            <s xml:id="echoid-s9537" xml:space="preserve">latus BC, palm. </s>
            <s xml:id="echoid-s9538" xml:space="preserve">12. </s>
            <s xml:id="echoid-s9539" xml:space="preserve">reperietur, ex demon-
              <lb/>
            ſtratis, angulus A, grad. </s>
            <s xml:id="echoid-s9540" xml:space="preserve">67. </s>
            <s xml:id="echoid-s9541" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s9542" xml:space="preserve">23. </s>
            <s xml:id="echoid-s9543" xml:space="preserve">& </s>
            <s xml:id="echoid-s9544" xml:space="preserve">angulus C, grad. </s>
            <s xml:id="echoid-s9545" xml:space="preserve">22. </s>
            <s xml:id="echoid-s9546" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s9547" xml:space="preserve">37. </s>
            <s xml:id="echoid-s9548" xml:space="preserve">latus vero
              <lb/>
            BC, partium 36. </s>
            <s xml:id="echoid-s9549" xml:space="preserve">qualium AB, eſt 15. </s>
            <s xml:id="echoid-s9550" xml:space="preserve">& </s>
            <s xml:id="echoid-s9551" xml:space="preserve">AC, 39. </s>
            <s xml:id="echoid-s9552" xml:space="preserve">Quare ſi fiat, vt latus BC,
              <lb/>
            inuentum partium 36. </s>
            <s xml:id="echoid-s9553" xml:space="preserve">ad idem BC, datum palm. </s>
            <s xml:id="echoid-s9554" xml:space="preserve">12. </s>
            <s xml:id="echoid-s9555" xml:space="preserve">ita tam AB, partium 15. </s>
            <s xml:id="echoid-s9556" xml:space="preserve">
              <lb/>
            quàm AC, partium 39. </s>
            <s xml:id="echoid-s9557" xml:space="preserve">ad aliud, inuenietur AB, palm. </s>
            <s xml:id="echoid-s9558" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9559" xml:space="preserve">& </s>
            <s xml:id="echoid-s9560" xml:space="preserve">AC, palm. </s>
            <s xml:id="echoid-s9561" xml:space="preserve">13. </s>
            <s xml:id="echoid-s9562" xml:space="preserve">
              <lb/>
            Datis ergo duobus lateribus trianguli rectanguli, duos angulos acutos effeci-
              <lb/>
            mus notos, &</s>
            <s xml:id="echoid-s9563" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9564" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s9565" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div813" type="section" level="1" n="457">
          <head xml:id="echoid-head489" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s9566" xml:space="preserve">_ABSOLVTVS_ iam eſt rectangulorum triangulorum calculus, ſequitur de
              <lb/>
            triangulis non rectangulis. </s>
            <s xml:id="echoid-s9567" xml:space="preserve">Sed prius quædam ad hanc rem neceſſaria demonſtranda
              <lb/>
            ſunt, quorum nonnulla plurimum etiam triangulis ſphæricis conducent.</s>
            <s xml:id="echoid-s9568" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div814" type="section" level="1" n="458">
          <head xml:id="echoid-head490" xml:space="preserve">THEOR. 2. PROPOS. 4.</head>
          <p>
            <s xml:id="echoid-s9569" xml:space="preserve">SI diameter circuli chordam quamlibet, eiusq́;
              <lb/>
            </s>
            <s xml:id="echoid-s9570" xml:space="preserve">
              <note position="left" xlink:label="note-322-02" xlink:href="note-322-02a" xml:space="preserve">Quam pro
                <lb/>
              portionem
                <lb/>
              habeãt duo
                <lb/>
              ſegmenta
                <lb/>
              cuiuſque
                <lb/>
              chordæ.</note>
            arcum ſecet in duas partes; </s>
            <s xml:id="echoid-s9571" xml:space="preserve">habebunt ſegmenta
              <lb/>
            chordæ eandem proportionem, quam ſinus ſeg-
              <lb/>
            mentorum arcus reſpondentium.</s>
            <s xml:id="echoid-s9572" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9573" xml:space="preserve">IN circulo ABCD, diameter AC, ſecet chordam BD, in E, eiuſq́ue ar-
              <lb/>
            cum BAD, in A, uel BCD, in C: </s>
            <s xml:id="echoid-s9574" xml:space="preserve">ducanturq́ue BF, DG, ad diametrum
              <lb/>
            AC, perpendiculares; </s>
            <s xml:id="echoid-s9575" xml:space="preserve">quarum BF, ſinus eſt arcus BA, uel BC: </s>
            <s xml:id="echoid-s9576" xml:space="preserve">& </s>
            <s xml:id="echoid-s9577" xml:space="preserve">DG, ſi-
              <lb/>
              <figure xlink:label="fig-322-01" xlink:href="fig-322-01a" number="170">
                <image file="322-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/322-01"/>
              </figure>
            nus arcus AD, uel CD. </s>
            <s xml:id="echoid-s9578" xml:space="preserve">Dico ita eſſe BE, ad ED,
              <lb/>
            ut BF, ad DG. </s>
            <s xml:id="echoid-s9579" xml:space="preserve">Quoniam enim in triangulis BE F,
              <lb/>
            DEG, anguli F, G, æquales ſunt, utpote recti: </s>
            <s xml:id="echoid-s9580" xml:space="preserve">Itẽ
              <lb/>
              <note position="left" xlink:label="note-322-03" xlink:href="note-322-03a" xml:space="preserve">15. primi.</note>
            anguli E, ad uerticem æquales; </s>
            <s xml:id="echoid-s9581" xml:space="preserve">æquiangula erunt
              <lb/>
              <note position="left" xlink:label="note-322-04" xlink:href="note-322-04a" xml:space="preserve">32. primi.</note>
            triangula BEF, DEG. </s>
            <s xml:id="echoid-s9582" xml:space="preserve">Quare erit, ut BE, ad BF,
              <lb/>
              <note position="left" xlink:label="note-322-05" xlink:href="note-322-05a" xml:space="preserve">4.ſexti.</note>
            ita ED, ad DG: </s>
            <s xml:id="echoid-s9583" xml:space="preserve">Et permutando, ut BE, ad ED,
              <lb/>
            ita BF, ad DG. </s>
            <s xml:id="echoid-s9584" xml:space="preserve">Si ergo diameter circuli chordam
              <lb/>
            quamlibet, eiusq́; </s>
            <s xml:id="echoid-s9585" xml:space="preserve">arcum ſecet in duas partes, &</s>
            <s xml:id="echoid-s9586" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9587" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s9588" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div817" type="section" level="1" n="459">
          <head xml:id="echoid-head491" xml:space="preserve">THEOR. 3. PROPOS. 5.</head>
          <p>
            <s xml:id="echoid-s9589" xml:space="preserve">SI in circulo chorda cuiuſlibet arcus ad vnam
              <lb/>
              <note position="left" xlink:label="note-322-06" xlink:href="note-322-06a" xml:space="preserve">Quã {pro}por
                <lb/>
              tionem ha
                <lb/>
              beat chor-
                <lb/>
              da circuli</note>
            partem producatur, conueniatq́; </s>
            <s xml:id="echoid-s9590" xml:space="preserve">cum </s>
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