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

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      <text xml:lang="la" type="free">
        <div xml:id="echoid-div562" type="section" level="1" n="262">
          <p>
            <s xml:id="echoid-s7923" xml:space="preserve">
              <pb o="195" file="207" n="207" rhead=""/>
            cunferentiam, angulus D, anguli I, duplus eſt; </s>
            <s xml:id="echoid-s7924" xml:space="preserve">erit quoque idem angulus D,
              <lb/>
              <note position="right" xlink:label="note-207-01" xlink:href="note-207-01a" xml:space="preserve">20. tertij.</note>
            anguli CAH, duplus: </s>
            <s xml:id="echoid-s7925" xml:space="preserve">ac proinde angulus CAH, ſemiſsis erit anguli D, qui
              <lb/>
            complementum eſt anguli DAC. </s>
            <s xml:id="echoid-s7926" xml:space="preserve">Cum ergo CH, tangens ſit anguli CAH,
              <lb/>
            ſitq́; </s>
            <s xml:id="echoid-s7927" xml:space="preserve">DA, recta rectæ DH, ex defin. </s>
            <s xml:id="echoid-s7928" xml:space="preserve">circuli, æqualis: </s>
            <s xml:id="echoid-s7929" xml:space="preserve">liquido conſtat, ſecan-
              <lb/>
            tem AD, arcus CE, æqualem eſſe tangenti CD, eiuſdem arcus, vnà cum CH,
              <lb/>
            tangente ſemiſsis complementi arcus CE, ſeu anguli CAD. </s>
            <s xml:id="echoid-s7930" xml:space="preserve">Quapropter Se-
              <lb/>
            cans cuiuſuis arcus æqualis eſt tangenti eiuſdẽ, &</s>
            <s xml:id="echoid-s7931" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7932" xml:space="preserve">Quod demonſtrandũ erat.</s>
            <s xml:id="echoid-s7933" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div566" type="section" level="1" n="263">
          <head xml:id="echoid-head290" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s7934" xml:space="preserve">EX preximis duobus theorematibus mirificum nobis compendium ſuppeditatur ad
              <lb/>
              <note position="right" xlink:label="note-207-02" xlink:href="note-207-02a" xml:space="preserve">Compédl @
                <lb/>
              mirificum
                <lb/>
              pro cóſtru-
                <lb/>
              ctione tabu
                <lb/>
              læ tam tan
                <lb/>
              gentium, q̃
                <lb/>
              ſecantium.</note>
            tabulam tam Tangentium, quàm Secantium conſtruendam. </s>
            <s xml:id="echoid-s7935" xml:space="preserve">Nam ſiper ea, quæ pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s7936" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7937" xml:space="preserve">eiuſq́; </s>
            <s xml:id="echoid-s7938" xml:space="preserve">ſcholio præcepim{us}, tangentes omnium arcuum per ſingula minuta ex-
              <lb/>
            tenſorum vſque ad ſemiſſem quadrantis, ſecantes vero omnium arcuum totius qua-
              <lb/>
            drantis inquiramus; </s>
            <s xml:id="echoid-s7939" xml:space="preserve">inueniemus earum beneficio per ſolam additionem tangentes
              <lb/>
            aliorum arcuum, vſque ad arcum grad 67. </s>
            <s xml:id="echoid-s7940" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7941" xml:space="preserve">30 ſi nimirum tangentem, & </s>
            <s xml:id="echoid-s7942" xml:space="preserve">ſecan-
              <lb/>
            tem cuiuſq; </s>
            <s xml:id="echoid-s7943" xml:space="preserve">arcus minoris ſemiſſe quadrantis, qui minuta numero paria contineat, in
              <lb/>
            vnam ſummá colligamus: </s>
            <s xml:id="echoid-s7944" xml:space="preserve">propterea quòd tangens cuiuſuis arc{us} maioris ſemiſſe qua-
              <lb/>
              <note position="right" xlink:label="note-207-03" xlink:href="note-207-03a" xml:space="preserve">19. huius.</note>
            drantis æqualis eſt tangenti, ac ſecanti arcus, qui duplus ſit exceſſus, quo datus arcus
              <lb/>
            ſemiſſem quadrantis ſuperat; </s>
            <s xml:id="echoid-s7945" xml:space="preserve">quales ſunt omnes arcus minutorũ parium vſq; </s>
            <s xml:id="echoid-s7946" xml:space="preserve">ad gra.
              <lb/>
            </s>
            <s xml:id="echoid-s7947" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7948" xml:space="preserve">vt arcus Min. </s>
            <s xml:id="echoid-s7949" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7950" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7951" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7952" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7953" xml:space="preserve">6. </s>
            <s xml:id="echoid-s7954" xml:space="preserve">&</s>
            <s xml:id="echoid-s7955" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7956" xml:space="preserve">Exempli cauſa. </s>
            <s xml:id="echoid-s7957" xml:space="preserve">ſi deſideretur tangens arcus
              <lb/>
            gra. </s>
            <s xml:id="echoid-s7958" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7959" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7960" xml:space="preserve">1 colligemus in vnam ſummam tangentem 5818. </s>
            <s xml:id="echoid-s7961" xml:space="preserve">& </s>
            <s xml:id="echoid-s7962" xml:space="preserve">ſecantẽ 10000002. </s>
            <s xml:id="echoid-s7963" xml:space="preserve">
              <lb/>
            arcus Min. </s>
            <s xml:id="echoid-s7964" xml:space="preserve">2 qui duplus eſt arcus Min. </s>
            <s xml:id="echoid-s7965" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7966" xml:space="preserve">quo datus arcus grad. </s>
            <s xml:id="echoid-s7967" xml:space="preserve">45 Min. </s>
            <s xml:id="echoid-s7968" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7969" xml:space="preserve">ſemißem quæ
              <lb/>
            drantis, hoc eſt, arcũ grad. </s>
            <s xml:id="echoid-s7970" xml:space="preserve">45 ſuperat. </s>
            <s xml:id="echoid-s7971" xml:space="preserve">Numerus enim cõſlatus 10005820. </s>
            <s xml:id="echoid-s7972" xml:space="preserve">dabit tã-
              <lb/>
            gentẽ propoſiti arcus grad. </s>
            <s xml:id="echoid-s7973" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7974" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7975" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7976" xml:space="preserve">qui ſemiſſem quadrantis ſuperat ſemiſſe arcus
              <lb/>
            Min. </s>
            <s xml:id="echoid-s7977" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7978" xml:space="preserve">vt propoſ. </s>
            <s xml:id="echoid-s7979" xml:space="preserve">19. </s>
            <s xml:id="echoid-s7980" xml:space="preserve">oſtenſum eſt. </s>
            <s xml:id="echoid-s7981" xml:space="preserve">Idẽ arcus grad. </s>
            <s xml:id="echoid-s7982" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7983" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7984" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7985" xml:space="preserve">cõponitur ex arcu Min. </s>
            <s xml:id="echoid-s7986" xml:space="preserve">
              <lb/>
            2. </s>
            <s xml:id="echoid-s7987" xml:space="preserve">& </s>
            <s xml:id="echoid-s7988" xml:space="preserve">ſemiſſe cõplemẽti eiuſdẽ arcus, quod cõplectitur grad. </s>
            <s xml:id="echoid-s7989" xml:space="preserve">89. </s>
            <s xml:id="echoid-s7990" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7991" xml:space="preserve">58. </s>
            <s xml:id="echoid-s7992" xml:space="preserve">hoc eſt, ex ar-
              <lb/>
            cu Min. </s>
            <s xml:id="echoid-s7993" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7994" xml:space="preserve">& </s>
            <s xml:id="echoid-s7995" xml:space="preserve">arcu grad. </s>
            <s xml:id="echoid-s7996" xml:space="preserve">44. </s>
            <s xml:id="echoid-s7997" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7998" xml:space="preserve">59 ac proinde, vt in ſcholio propoſ. </s>
            <s xml:id="echoid-s7999" xml:space="preserve">19. </s>
            <s xml:id="echoid-s8000" xml:space="preserve">demonſtraui-
              <lb/>
            mus, numerus 10005820. </s>
            <s xml:id="echoid-s8001" xml:space="preserve">conſlatus ex tangente, & </s>
            <s xml:id="echoid-s8002" xml:space="preserve">ſecante arcus Min. </s>
            <s xml:id="echoid-s8003" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8004" xml:space="preserve">dabit tan-
              <lb/>
            gentem dicti arcus compoſiti ex arcu Min. </s>
            <s xml:id="echoid-s8005" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8006" xml:space="preserve">& </s>
            <s xml:id="echoid-s8007" xml:space="preserve">ſemiſſe complementi eiuſdem.</s>
            <s xml:id="echoid-s8008" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8009" xml:space="preserve">EADEM ratione tangens cuiuſuis arcus mauris ſemiſſe quadrantis compone-
              <lb/>
            tur ex tangẽte, & </s>
            <s xml:id="echoid-s8010" xml:space="preserve">ſecante arcus, qui duplus ſit exceſſus, quo arcus ille ſemiſſem qua-
              <lb/>
            drantis ſuperat. </s>
            <s xml:id="echoid-s8011" xml:space="preserve">Item ſi in vnam ſummã colligantur tangens, & </s>
            <s xml:id="echoid-s8012" xml:space="preserve">ſecans cuiuſuis ar-
              <lb/>
            cus minoris ſemiſſe quadrátis, conſlabitur tàgens arcus cõpoſiti ex illo arcu, & </s>
            <s xml:id="echoid-s8013" xml:space="preserve">ſemiſ
              <lb/>
            ſe complementi eiuſdem. </s>
            <s xml:id="echoid-s8014" xml:space="preserve">Vt tangens 14352451. </s>
            <s xml:id="echoid-s8015" xml:space="preserve">arcus grad. </s>
            <s xml:id="echoid-s8016" xml:space="preserve">55 Min. </s>
            <s xml:id="echoid-s8017" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8018" xml:space="preserve">compoſita eſt
              <lb/>
            ex tangente 3692500. </s>
            <s xml:id="echoid-s8019" xml:space="preserve">& </s>
            <s xml:id="echoid-s8020" xml:space="preserve">ſecante 10659951. </s>
            <s xml:id="echoid-s8021" xml:space="preserve">arcus grad. </s>
            <s xml:id="echoid-s8022" xml:space="preserve">20 Min. </s>
            <s xml:id="echoid-s8023" xml:space="preserve">16. </s>
            <s xml:id="echoid-s8024" xml:space="preserve">qui duplus eſt
              <lb/>
            arcus grad. </s>
            <s xml:id="echoid-s8025" xml:space="preserve">10. </s>
            <s xml:id="echoid-s8026" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8027" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8028" xml:space="preserve">quo datus arcus grad. </s>
            <s xml:id="echoid-s8029" xml:space="preserve">55 Min. </s>
            <s xml:id="echoid-s8030" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8031" xml:space="preserve">quadrantis ſemiſſem ſuperat.
              <lb/>
            </s>
            <s xml:id="echoid-s8032" xml:space="preserve">Item ſi tangens 3692500. </s>
            <s xml:id="echoid-s8033" xml:space="preserve">& </s>
            <s xml:id="echoid-s8034" xml:space="preserve">ſecans 10659951. </s>
            <s xml:id="echoid-s8035" xml:space="preserve">arcus grad. </s>
            <s xml:id="echoid-s8036" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8037" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8038" xml:space="preserve">16. </s>
            <s xml:id="echoid-s8039" xml:space="preserve">in vnam
              <lb/>
            ſummam colligantur, conſlabitur tangens 14352451. </s>
            <s xml:id="echoid-s8040" xml:space="preserve">arcus grad 55. </s>
            <s xml:id="echoid-s8041" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8042" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8043" xml:space="preserve">comſoſi-
              <lb/>
            ti ex illo arcu grad. </s>
            <s xml:id="echoid-s8044" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8045" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8046" xml:space="preserve">16. </s>
            <s xml:id="echoid-s8047" xml:space="preserve">& </s>
            <s xml:id="echoid-s8048" xml:space="preserve">ſemiſſe complementi eiuſdem, nempe ex arcu grad. </s>
            <s xml:id="echoid-s8049" xml:space="preserve">
              <lb/>
            34. </s>
            <s xml:id="echoid-s8050" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8051" xml:space="preserve">52. </s>
            <s xml:id="echoid-s8052" xml:space="preserve">cum complementũ arcus grad. </s>
            <s xml:id="echoid-s8053" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8054" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8055" xml:space="preserve">16. </s>
            <s xml:id="echoid-s8056" xml:space="preserve">cõprehendat grad. </s>
            <s xml:id="echoid-s8057" xml:space="preserve">69. </s>
            <s xml:id="echoid-s8058" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8059" xml:space="preserve">44.</s>
            <s xml:id="echoid-s8060" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8061" xml:space="preserve">
              <emph style="sc">ITAQVe</emph>
            propoſito arcu quocunque, qui maior ſit quadrantis dimidio, ſi ex
              <lb/>
            eo detrahatur ſemiſsis quadrantis, id eſt, arcus grad. </s>
            <s xml:id="echoid-s8062" xml:space="preserve">45. </s>
            <s xml:id="echoid-s8063" xml:space="preserve">& </s>
            <s xml:id="echoid-s8064" xml:space="preserve">reliqui arcus ſumatur
              <lb/>
            duplus, component tangens & </s>
            <s xml:id="echoid-s8065" xml:space="preserve">ſecans huius dupli arcus ſumpti tangentem propoſiti
              <lb/>
            illius arcus. </s>
            <s xml:id="echoid-s8066" xml:space="preserve">Vt ſi quæratur tangens arcus grad. </s>
            <s xml:id="echoid-s8067" xml:space="preserve">55. </s>
            <s xml:id="echoid-s8068" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8069" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8070" xml:space="preserve">detrahendi erunt grad.
              <lb/>
            </s>
            <s xml:id="echoid-s8071" xml:space="preserve">45. </s>
            <s xml:id="echoid-s8072" xml:space="preserve">ex eo, & </s>
            <s xml:id="echoid-s8073" xml:space="preserve">reliqui arcus grad. </s>
            <s xml:id="echoid-s8074" xml:space="preserve">10. </s>
            <s xml:id="echoid-s8075" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s8076" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8077" xml:space="preserve">ſumendus duplus arcus grad. </s>
            <s xml:id="echoid-s8078" xml:space="preserve">20. </s>
            <s xml:id="echoid-s8079" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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