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

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        <div xml:id="echoid-div553" type="section" level="1" n="259">
          <p style="it">
            <s xml:id="echoid-s7834" xml:space="preserve">
              <pb o="193" file="205" n="205" rhead=""/>
            100000000000000. </s>
            <s xml:id="echoid-s7835" xml:space="preserve">per sinus complementorum omnium arcuum, initio facto à
              <lb/>
              <note position="right" xlink:label="note-205-01" xlink:href="note-205-01a" xml:space="preserve">Sola diui-
                <lb/>
              ſione eiuſ-
                <lb/>
              dé ſemper
                <lb/>
              numeri p ſi
                <lb/>
              nus omnes
                <lb/>
              ſecantes in
                <lb/>
              ueniuntur.</note>
            principto Quadrantis, omnium arcuum ſecantes eruuntur, vt ex demonſtratis liquet.
              <lb/>
            </s>
            <s xml:id="echoid-s7836" xml:space="preserve">Ex quo facilima erit conſtructio tabulæ ſecantium.</s>
            <s xml:id="echoid-s7837" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div556" type="section" level="1" n="260">
          <head xml:id="echoid-head287" xml:space="preserve">THEOR 11. PROPOS. 19.</head>
          <p>
            <s xml:id="echoid-s7838" xml:space="preserve">TANGENS cuiuſuis arcus, qui ſemiſſe qua-
              <lb/>
              <note position="right" xlink:label="note-205-02" xlink:href="note-205-02a" xml:space="preserve">Tangens ar
                <lb/>
              cus maioris
                <lb/>
              ſemiſle qua
                <lb/>
              diantis, cui
                <lb/>
              tangenti, &
                <lb/>
              ſecanti ſi-
                <lb/>
              mul ſit æ-
                <lb/>
              qualis.</note>
            drantis maior ſit, æqualis eſt tangenti & </s>
            <s xml:id="echoid-s7839" xml:space="preserve">ſecanti ſi-
              <lb/>
            mul arcus, qui duplus ſit exceſſus, quo datus ar-
              <lb/>
            cus ſemiſſem quadrantis ſuperat.</s>
            <s xml:id="echoid-s7840" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7841" xml:space="preserve">IN quadrante ABC, ſit CG, tangens arcus CF, qui ſemiſſe quadrantis
              <lb/>
            maior ſit, inter ſemidiametrum AC, & </s>
            <s xml:id="echoid-s7842" xml:space="preserve">ſecantem AG, eiuſdem arcus CF, com
              <lb/>
            prehenſa. </s>
            <s xml:id="echoid-s7843" xml:space="preserve">Dico CG, æqualem eſſe tangenti, & </s>
            <s xml:id="echoid-s7844" xml:space="preserve">ſecanti ſimul arcus, qui duplus
              <lb/>
            ſit exceſſus, quo arcus CF, ſemiſſem quadrantis
              <lb/>
              <figure xlink:label="fig-205-01" xlink:href="fig-205-01a" number="152">
                <image file="205-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/205-01"/>
              </figure>
            ſuperat. </s>
            <s xml:id="echoid-s7845" xml:space="preserve">Sumpto enim arcu FD, ipſi FB, æquali,
              <lb/>
            ducatur recta AD, extendaturq́; </s>
            <s xml:id="echoid-s7846" xml:space="preserve">vſque ad E. </s>
            <s xml:id="echoid-s7847" xml:space="preserve">Et
              <lb/>
            quoniam anguli BAF, FAE, ob æquales arcus
              <lb/>
              <note position="right" xlink:label="note-205-03" xlink:href="note-205-03a" xml:space="preserve">27. tertij.</note>
            BF, DF, æquales ſunt: </s>
            <s xml:id="echoid-s7848" xml:space="preserve">Et angulo BAF, æqualis
              <lb/>
            eſt alternus angulus G; </s>
            <s xml:id="echoid-s7849" xml:space="preserve">erit quoq; </s>
            <s xml:id="echoid-s7850" xml:space="preserve">idem angulus
              <lb/>
              <note position="right" xlink:label="note-205-04" xlink:href="note-205-04a" xml:space="preserve">29. primi.</note>
            G, angulo GAE, æqualis. </s>
            <s xml:id="echoid-s7851" xml:space="preserve">Quare rectæ EG, EA,
              <lb/>
              <note position="right" xlink:label="note-205-05" xlink:href="note-205-05a" xml:space="preserve">6. primi.</note>
            æquales ſunt: </s>
            <s xml:id="echoid-s7852" xml:space="preserve">ac propterea, addita communi CE,
              <lb/>
            erit CG, tota tangens arcus CF, duabus CE, & </s>
            <s xml:id="echoid-s7853" xml:space="preserve">
              <lb/>
            AE, hoc eſt, tangenti, & </s>
            <s xml:id="echoid-s7854" xml:space="preserve">ſecanti arcus CD, ſimul
              <lb/>
            æqualis. </s>
            <s xml:id="echoid-s7855" xml:space="preserve">Dico iam arcum CD, duplum eſſe exceſ-
              <lb/>
            ſus quo arcus CF, propoſitus ſemiſſem quadrãtis ſuperat. </s>
            <s xml:id="echoid-s7856" xml:space="preserve">Producto enim ar-
              <lb/>
            cu quadrantis ad partes B, ſumptoq́; </s>
            <s xml:id="echoid-s7857" xml:space="preserve">arcu BH, æquali ipſi CD, cum & </s>
            <s xml:id="echoid-s7858" xml:space="preserve">arcus
              <lb/>
            FB, arcui FD, ſit æqualis, erit totus arcus FH, toti arcui CF, æqualis, ac pro-
              <lb/>
            inde arcus CH, duplus erit arcus CF. </s>
            <s xml:id="echoid-s7859" xml:space="preserve">Quoniam vero arcus CH, quadrantem
              <lb/>
            CB, ſuperat arcu BH, hoc eſt, arcu CD; </s>
            <s xml:id="echoid-s7860" xml:space="preserve">ſuperabit CF, ſemiſsis arcus CH, ſe
              <lb/>
              <note position="right" xlink:label="note-205-06" xlink:href="note-205-06a" xml:space="preserve">7. huius.</note>
            miſsem quadrantis CB, ſemiſse exceſſus CD. </s>
            <s xml:id="echoid-s7861" xml:space="preserve">Arcus igitur CD, duplus eſt ex-
              <lb/>
            ceſſus, quo datus arcus CF, ſemiſſem quadrantis ſuperat. </s>
            <s xml:id="echoid-s7862" xml:space="preserve">Eſt auté oſtenſum,
              <lb/>
            CG, tangentem arcus CF, æqualem eſſe tangenti CE, & </s>
            <s xml:id="echoid-s7863" xml:space="preserve">ſecanti AE, ſimul ar-
              <lb/>
            cus CD. </s>
            <s xml:id="echoid-s7864" xml:space="preserve">Igitur tangens cuiuſuis arcus, qui ſemiſſe quadrantis maior ſit, æqua
              <lb/>
            lis eſt tangenti, & </s>
            <s xml:id="echoid-s7865" xml:space="preserve">ſecanti ſimul arcus, qui duplus ſit exceſſus, quo datus ar-
              <lb/>
            cus ſemiſſem quadrantis ſuperat. </s>
            <s xml:id="echoid-s7866" xml:space="preserve">Quod oſtendendum erat.</s>
            <s xml:id="echoid-s7867" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div559" type="section" level="1" n="261">
          <head xml:id="echoid-head288" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s7868" xml:space="preserve">HANC propoſitionem nonnulli ita proponunt.</s>
            <s xml:id="echoid-s7869" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7870" xml:space="preserve">SECANS cuiuſuis arcus vna cum tangente eiuſdem æqualis eſt
              <lb/>
              <note position="right" xlink:label="note-205-07" xlink:href="note-205-07a" xml:space="preserve">Secás @@ tá-
                <lb/>
              gens eiuſ-
                <lb/>
              dem arcus
                <lb/>
              cui tágenti
                <lb/>
              ſimul ęqua
                <lb/>
              les ſint.</note>
            tangenti arcus compoſiti ex dato arcu, & </s>
            <s xml:id="echoid-s7871" xml:space="preserve">ſemiſſe complementi
              <lb/>
            eiuſdem.</s>
            <s xml:id="echoid-s7872" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7873" xml:space="preserve">N A M in eadem figura ſit AE, ſecans, & </s>
            <s xml:id="echoid-s7874" xml:space="preserve">
              <emph style="sc">Ce</emph>
            , tangens eiuſdem arcus CD. </s>
            <s xml:id="echoid-s7875" xml:space="preserve"/>
          </p>
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