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

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        <div xml:id="echoid-div784" type="section" level="1" n="454">
          <p style="it">
            <s xml:id="echoid-s8956" xml:space="preserve">
              <pb o="303" file="315" n="315" rhead=""/>
            guli ACB, duplicatus dabit latus AB, in partibus ſemidiametri AD,
              <lb/>
            &</s>
            <s xml:id="echoid-s8957" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8958" xml:space="preserve">Latus autem recto angulo oppoſitum perpetuò eſt diameter circuli
              <lb/>
            circunſcripti triangulo. </s>
            <s xml:id="echoid-s8959" xml:space="preserve">quare ſi ſemidiameter, ſinus ve totus duplicetur,
              <lb/>
            cognitum fiet ipſum latus. </s>
            <s xml:id="echoid-s8960" xml:space="preserve">Latus deniq; </s>
            <s xml:id="echoid-s8961" xml:space="preserve">obtuſo angulo oppoſitum habe-
              <lb/>
            bitur, ſi vterq; </s>
            <s xml:id="echoid-s8962" xml:space="preserve">angulorum acutorum duplicetur, & </s>
            <s xml:id="echoid-s8963" xml:space="preserve">duplicatorum ſemiſ-
              <lb/>
            ſis accipiatur. </s>
            <s xml:id="echoid-s8964" xml:space="preserve">Nam ſinus huius ſemiſsis duplicatus illico latus oſtendet
              <lb/>
              <note position="right" xlink:label="note-315-01" xlink:href="note-315-01a" xml:space="preserve">20. tertlj.</note>
            notum. </s>
            <s xml:id="echoid-s8965" xml:space="preserve">Anguli namq; </s>
            <s xml:id="echoid-s8966" xml:space="preserve">ADB, ADC, dupli ſunt acutorum angulorum
              <lb/>
            ACB, ABC: </s>
            <s xml:id="echoid-s8967" xml:space="preserve">quibus quidem duplis angulis totus arcus BAC, de-
              <lb/>
            betur, &</s>
            <s xml:id="echoid-s8968" xml:space="preserve">c.</s>
            <s xml:id="echoid-s8969" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8970" xml:space="preserve">_IMMO_ vero ſinon dentur anguli, ſed eorum tantum proportiones, cogneſce-
              <lb/>
              <note position="right" xlink:label="note-315-02" xlink:href="note-315-02a" xml:space="preserve">Quomodo
                <lb/>
              ex datis {pro}-
                <lb/>
              portionibꝰ
                <lb/>
              omniũ an-
                <lb/>
              gulotũ triã
                <lb/>
              guli cogno
                <lb/>
              ſcantur ipſi
                <lb/>
              anguli.</note>
            mus nihilominus proportiones laterum, ſi prius ex angulorum proportionibus datis
              <lb/>
            eorundem magnitudines inueſtigemus, hocmodo. </s>
            <s xml:id="echoid-s8971" xml:space="preserve">In primo triangulo prioris figur æ
              <lb/>
            huius ſcholij ponatur proportio anguli _C,_ ad angulum _B,_ eadem, quæ _12._ </s>
            <s xml:id="echoid-s8972" xml:space="preserve">ad _10._ </s>
            <s xml:id="echoid-s8973" xml:space="preserve">& </s>
            <s xml:id="echoid-s8974" xml:space="preserve">
              <lb/>
            anguli _B,_ ad angulum _A,_ quæ _20._ </s>
            <s xml:id="echoid-s8975" xml:space="preserve">ad _28._ </s>
            <s xml:id="echoid-s8976" xml:space="preserve">quæ duæ proportiones notæ ſatis ſunt, e-
              <lb/>
            tiamſi proportio anguli _A,_ ad angulum _C,_ ignota ſit. </s>
            <s xml:id="echoid-s8977" xml:space="preserve">Inuentis autem minimis nu-
              <lb/>
            meris _6. </s>
            <s xml:id="echoid-s8978" xml:space="preserve">5._ </s>
            <s xml:id="echoid-s8979" xml:space="preserve">qui eandem proportio-
              <lb/>
              <figure xlink:label="fig-315-01" xlink:href="fig-315-01a" number="165">
                <image file="315-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/315-01"/>
              </figure>
              <note position="right" xlink:label="note-315-03" xlink:href="note-315-03a" xml:space="preserve">35. ſeptimi</note>
            nem habeant, quam anguli _C, B,_ hoc
              <lb/>
            eſt, quam numeri _12. </s>
            <s xml:id="echoid-s8980" xml:space="preserve">10._ </s>
            <s xml:id="echoid-s8981" xml:space="preserve">ſihi mini-
              <lb/>
            mi non ſint; </s>
            <s xml:id="echoid-s8982" xml:space="preserve">Item minimis _5. </s>
            <s xml:id="echoid-s8983" xml:space="preserve">7._ </s>
            <s xml:id="echoid-s8984" xml:space="preserve">ean-
              <lb/>
            dem proportionem habentibus, quam
              <lb/>
            anguli _B, A,_ ſiue numeri _20. </s>
            <s xml:id="echoid-s8985" xml:space="preserve">28._ </s>
            <s xml:id="echoid-s8986" xml:space="preserve">ſu-
              <lb/>
            memus tres hoſce numeros deinceps
              <lb/>
            minimos _6. </s>
            <s xml:id="echoid-s8987" xml:space="preserve">5. </s>
            <s xml:id="echoid-s8988" xml:space="preserve">7._ </s>
            <s xml:id="echoid-s8989" xml:space="preserve">in proportionibus
              <lb/>
            numerorum minimorum _6. </s>
            <s xml:id="echoid-s8990" xml:space="preserve">5._ </s>
            <s xml:id="echoid-s8991" xml:space="preserve">& </s>
            <s xml:id="echoid-s8992" xml:space="preserve">_5 7._
              <lb/>
            </s>
            <s xml:id="echoid-s8993" xml:space="preserve">qui ſi non eſſent deinceps minimi, in-
              <lb/>
            quirendi eſſent tres minimi, per ea, quæ ab Euclide demonſtrata ſunt lib. </s>
            <s xml:id="echoid-s8994" xml:space="preserve">_8._ </s>
            <s xml:id="echoid-s8995" xml:space="preserve">Erit
              <lb/>
              <note position="right" xlink:label="note-315-04" xlink:href="note-315-04a" xml:space="preserve">4. octaui.</note>
            ergo angulus _C,_ vt _6. </s>
            <s xml:id="echoid-s8996" xml:space="preserve">B,_ vt _5._ </s>
            <s xml:id="echoid-s8997" xml:space="preserve">& </s>
            <s xml:id="echoid-s8998" xml:space="preserve">_A,_ vt _7._ </s>
            <s xml:id="echoid-s8999" xml:space="preserve">quos in gradibus per regulam Societatum
              <lb/>
            ita notos efficiemus. </s>
            <s xml:id="echoid-s9000" xml:space="preserve">Collectis numeris _6. </s>
            <s xml:id="echoid-s9001" xml:space="preserve">5. </s>
            <s xml:id="echoid-s9002" xml:space="preserve">7._ </s>
            <s xml:id="echoid-s9003" xml:space="preserve">in vnam ſummam _18._ </s>
            <s xml:id="echoid-s9004" xml:space="preserve">dicemus per
              <lb/>
            auream regulam. </s>
            <s xml:id="echoid-s9005" xml:space="preserve">Si _18._ </s>
            <s xml:id="echoid-s9006" xml:space="preserve">dant grad. </s>
            <s xml:id="echoid-s9007" xml:space="preserve">_180._ </s>
            <s xml:id="echoid-s9008" xml:space="preserve">(tot enim gradibus omnes tres anguli, hoc
              <lb/>
            eſt, duo recti, æquiualent.) </s>
            <s xml:id="echoid-s9009" xml:space="preserve">quid dabunt 6? </s>
            <s xml:id="echoid-s9010" xml:space="preserve">quid 5? </s>
            <s xml:id="echoid-s9011" xml:space="preserve">& </s>
            <s xml:id="echoid-s9012" xml:space="preserve">quid 7? </s>
            <s xml:id="echoid-s9013" xml:space="preserve">vt hic vides.</s>
            <s xml:id="echoid-s9014" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9015" xml:space="preserve">18. </s>
            <s xml:id="echoid-s9016" xml:space="preserve">180. </s>
            <s xml:id="echoid-s9017" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s9018" xml:space="preserve">{6? </s>
            <s xml:id="echoid-s9019" xml:space="preserve">\\ 5? </s>
            <s xml:id="echoid-s9020" xml:space="preserve">\\ 7?</s>
            <s xml:id="echoid-s9021" xml:space="preserve">} fiunt {60. </s>
            <s xml:id="echoid-s9022" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s9023" xml:space="preserve">\\ 50. </s>
            <s xml:id="echoid-s9024" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s9025" xml:space="preserve">\\ 70. </s>
            <s xml:id="echoid-s9026" xml:space="preserve">gr.</s>
            <s xml:id="echoid-s9027" xml:space="preserve">} pro angulo {C. </s>
            <s xml:id="echoid-s9028" xml:space="preserve">\\ B. </s>
            <s xml:id="echoid-s9029" xml:space="preserve">\\ A.
              <lb/>
            </s>
            <s xml:id="echoid-s9030" xml:space="preserve">Inueniemusq́; </s>
            <s xml:id="echoid-s9031" xml:space="preserve">angulum C, grad. </s>
            <s xml:id="echoid-s9032" xml:space="preserve">60. </s>
            <s xml:id="echoid-s9033" xml:space="preserve">_
              <emph style="sc">B</emph>
            ,_ grad. </s>
            <s xml:id="echoid-s9034" xml:space="preserve">_50._ </s>
            <s xml:id="echoid-s9035" xml:space="preserve">& </s>
            <s xml:id="echoid-s9036" xml:space="preserve">_A,_ grad. </s>
            <s xml:id="echoid-s9037" xml:space="preserve">_70._ </s>
            <s xml:id="echoid-s9038" xml:space="preserve">Quòd ſi duæ no-
              <lb/>
              <note position="right" xlink:label="note-315-05" xlink:href="note-315-05a" xml:space="preserve">Quãdo pro
                <lb/>
              poitiones
                <lb/>
              angulorũ
                <lb/>
              notæ non
                <lb/>
              ſunt conti-
                <lb/>
              nuatę, quid
                <lb/>
              agendum.</note>
            tæ proportiones angulorum non ſint continuatæ, vt in dato exemplo, continuandæ
              <lb/>
            erunt. </s>
            <s xml:id="echoid-s9039" xml:space="preserve">Vt ſi dicat quis. </s>
            <s xml:id="echoid-s9040" xml:space="preserve">Proportio anguli _C,_ ad angulum _B,_ eſt vt _12._ </s>
            <s xml:id="echoid-s9041" xml:space="preserve">ad _10._ </s>
            <s xml:id="echoid-s9042" xml:space="preserve">& </s>
            <s xml:id="echoid-s9043" xml:space="preserve">pro-
              <lb/>
            portio anguli _A,_ ad angulum _
              <emph style="sc">B</emph>
            ,_ vt _28._ </s>
            <s xml:id="echoid-s9044" xml:space="preserve">ad _20._ </s>
            <s xml:id="echoid-s9045" xml:space="preserve">vbi vides, eundem angulum _C,_ in
              <lb/>
            vtraq; </s>
            <s xml:id="echoid-s9046" xml:space="preserve">proportione eſſe conſequens: </s>
            <s xml:id="echoid-s9047" xml:space="preserve">continuabimus illas, ſi dicamus, proportionem
              <lb/>
            _C,_ ad _
              <emph style="sc">B</emph>
            ,_ eſſe vt _12._ </s>
            <s xml:id="echoid-s9048" xml:space="preserve">ad _10._ </s>
            <s xml:id="echoid-s9049" xml:space="preserve">& </s>
            <s xml:id="echoid-s9050" xml:space="preserve">_
              <emph style="sc">B</emph>
            ,_ ad _A,_ vt _20._ </s>
            <s xml:id="echoid-s9051" xml:space="preserve">ad _28._ </s>
            <s xml:id="echoid-s9052" xml:space="preserve">Aut ſi quis dicat. </s>
            <s xml:id="echoid-s9053" xml:space="preserve">Proportio an-
              <lb/>
            guli _C,_ ad _
              <emph style="sc">B</emph>
            ,_ eſt vt _12._ </s>
            <s xml:id="echoid-s9054" xml:space="preserve">ad _10._ </s>
            <s xml:id="echoid-s9055" xml:space="preserve">proportio autem _A,_ ad _C,_ eſt vt _28._ </s>
            <s xml:id="echoid-s9056" xml:space="preserve">ad _24._ </s>
            <s xml:id="echoid-s9057" xml:space="preserve">continua-
              <lb/>
            bimus eas, ponendo proportionem _A,_ ad _C,_ vt _28._ </s>
            <s xml:id="echoid-s9058" xml:space="preserve">ad _24._ </s>
            <s xml:id="echoid-s9059" xml:space="preserve">& </s>
            <s xml:id="echoid-s9060" xml:space="preserve">_C,_ ad _
              <emph style="sc">B</emph>
            ,_ vt _12._ </s>
            <s xml:id="echoid-s9061" xml:space="preserve">ad _10._
              <lb/>
            </s>
            <s xml:id="echoid-s9062" xml:space="preserve">Aut deniq; </s>
            <s xml:id="echoid-s9063" xml:space="preserve">ſi quis dicat. </s>
            <s xml:id="echoid-s9064" xml:space="preserve">Proportio _C,_ ad _
              <emph style="sc">B</emph>
            ,_ eſt vt _12._ </s>
            <s xml:id="echoid-s9065" xml:space="preserve">ad _10._ </s>
            <s xml:id="echoid-s9066" xml:space="preserve">& </s>
            <s xml:id="echoid-s9067" xml:space="preserve">_C,_ ad _A,_ vt _24._ </s>
            <s xml:id="echoid-s9068" xml:space="preserve"/>
          </p>
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      </text>
    </echo>
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