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

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          <p>
            <s xml:id="echoid-s6853" xml:space="preserve">
              <pb o="182" file="194" n="194" rhead=""/>
            proportionalis eſt inter AC, FC; </s>
            <s xml:id="echoid-s6854" xml:space="preserve">erit rectangulum ſub AC, FC, æquale
              <lb/>
              <note position="left" xlink:label="note-194-01" xlink:href="note-194-01a" xml:space="preserve">15.ſexti.</note>
            quadrato rectæ CD. </s>
            <s xml:id="echoid-s6855" xml:space="preserve">Cum ergo illud notũ ſit, ob notas AC, FC; </s>
            <s xml:id="echoid-s6856" xml:space="preserve">erit & </s>
            <s xml:id="echoid-s6857" xml:space="preserve">qua-
              <lb/>
            dratum ex DC, notum: </s>
            <s xml:id="echoid-s6858" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s6859" xml:space="preserve">adeo radix eius quadrata rectam DC, efficiet
              <lb/>
            notam. </s>
            <s xml:id="echoid-s6860" xml:space="preserve">Quare ex data chorda cuiuſuis arcus chordam ſemiſsis illius arcus in-
              <lb/>
            uenimus. </s>
            <s xml:id="echoid-s6861" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s6862" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div520" type="section" level="1" n="252">
          <head xml:id="echoid-head279" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s6863" xml:space="preserve">ITAQVE, ſi per propoſ. </s>
            <s xml:id="echoid-s6864" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6865" xml:space="preserve">inueniatur chorda arcus, qui cum arcu datæ chordæ ſemicir-
              <lb/>
              <note position="left" xlink:label="note-194-02" xlink:href="note-194-02a" xml:space="preserve">Praxis.</note>
            culum conſicit; </s>
            <s xml:id="echoid-s6866" xml:space="preserve">inuentæ autem huius chordæ dimidium ex ſemidiametro detrahatur, & </s>
            <s xml:id="echoid-s6867" xml:space="preserve">
              <lb/>
            reliquus numerus in diametrum multiplicetur, dabit radix quadrata huius producti chor-
              <lb/>
            dam ſemiſsis illius arcus, cuius chorda data eſt. </s>
            <s xml:id="echoid-s6868" xml:space="preserve">Vel ſi reliqui illius numeri quadratum iun
              <lb/>
            gatur quadrato ſemiſsis chordæ datæ, componetur numerus, cuius radix quadrata chordam
              <lb/>
            quæſitam exhibebit cognitam. </s>
            <s xml:id="echoid-s6869" xml:space="preserve">Quæ quidem operatio facile colligitur ex figura, & </s>
            <s xml:id="echoid-s6870" xml:space="preserve">priori de-
              <lb/>
            mon ſtratione huius propoſ.</s>
            <s xml:id="echoid-s6871" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6872" xml:space="preserve">ITEM ſi per propoſ. </s>
            <s xml:id="echoid-s6873" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6874" xml:space="preserve">reperiatur chorda arcus, qui cum arcu datæ chordæ ſemicircu-
              <lb/>
            lum conſicit; </s>
            <s xml:id="echoid-s6875" xml:space="preserve">inuenta autem hæc chorda ex dia metro detrabatur, & </s>
            <s xml:id="echoid-s6876" xml:space="preserve">reliqui numeri dimi-
              <lb/>
            dium in diametrum multiplicetur, dabit radix quadrata huius producti chordam ſemiſſis
              <lb/>
            illius arcus, cuius chorda data eſt. </s>
            <s xml:id="echoid-s6877" xml:space="preserve">Vt perſpicuum eſt ex figura, & </s>
            <s xml:id="echoid-s6878" xml:space="preserve">poſteriori demonſtra-
              <lb/>
            tione huius propoſ.</s>
            <s xml:id="echoid-s6879" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div522" type="section" level="1" n="253">
          <head xml:id="echoid-head280" xml:space="preserve">PROBL. 8. PROPOS. 16.</head>
          <p>
            <s xml:id="echoid-s6880" xml:space="preserve">CHORDAS omnium arcuum ſemicirculi
              <lb/>
              <note position="left" xlink:label="note-194-03" xlink:href="note-194-03a" xml:space="preserve">Qua ratio-
                <lb/>
              tione om-
                <lb/>
              niu arcuũ
                <lb/>
              chordæ ſup
                <lb/>
              putentur.</note>
            ſeſe ordine ſuperantium vno Minuto, in partibus
              <lb/>
            diametri in quotius particulas diſtributæ, ſuppu-
              <lb/>
            tare.</s>
            <s xml:id="echoid-s6881" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6882" xml:space="preserve">STATVAMVS, chordas omnium arcuum ſupputandas eſſe reſpectu
              <lb/>
            diametri in partes 200000. </s>
            <s xml:id="echoid-s6883" xml:space="preserve">diſtributæ. </s>
            <s xml:id="echoid-s6884" xml:space="preserve">Quod vt fiat accuratius, ponenda erit
              <lb/>
            in ſupputationibus diameter partium 20000000. </s>
            <s xml:id="echoid-s6885" xml:space="preserve">Ita enim fiet, vt abiectis dua-
              <lb/>
            bus primis figuris ad dexteram ex ſingulis chordis inuentis, relinquãtur chor-
              <lb/>
            dæ magis exquiſitę reſpectu diametri partium 200000. </s>
            <s xml:id="echoid-s6886" xml:space="preserve">quemadmodum ad ini-
              <lb/>
            tium propoſ. </s>
            <s xml:id="echoid-s6887" xml:space="preserve">9. </s>
            <s xml:id="echoid-s6888" xml:space="preserve">de Sinubus docuimus, vbi etiam addidimus, quot particula-
              <lb/>
            rum ſinus totus aſſumi debeat in ſupputatione, ſi ſinus totus in tabula plu-
              <lb/>
            rium particularum deſideretur. </s>
            <s xml:id="echoid-s6889" xml:space="preserve">Quod etiam de tota diametro hic intelligi
              <lb/>
            debet, ſtatuendo ſemper diametrum duplo plurium particularum in ſuppu-
              <lb/>
            titione, quam ſinum totum ibi conſtituimus.</s>
            <s xml:id="echoid-s6890" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6891" xml:space="preserve">PRIMVM ergo omnium inuentæ ſunt in propoſ. </s>
            <s xml:id="echoid-s6892" xml:space="preserve">12. </s>
            <s xml:id="echoid-s6893" xml:space="preserve">chordæ arcuum
              <lb/>
            grad. </s>
            <s xml:id="echoid-s6894" xml:space="preserve">36. </s>
            <s xml:id="echoid-s6895" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s6896" xml:space="preserve">60. </s>
            <s xml:id="echoid-s6897" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s6898" xml:space="preserve">72. </s>
            <s xml:id="echoid-s6899" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s6900" xml:space="preserve">90. </s>
            <s xml:id="echoid-s6901" xml:space="preserve">& </s>
            <s xml:id="echoid-s6902" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s6903" xml:space="preserve">120. </s>
            <s xml:id="echoid-s6904" xml:space="preserve">nempe latera decagoni, he-
              <lb/>
            xagoni, pentagoni, quadrati, & </s>
            <s xml:id="echoid-s6905" xml:space="preserve">trianguli æquilateri, partium 6180340.
              <lb/>
            </s>
            <s xml:id="echoid-s6906" xml:space="preserve">10000000. </s>
            <s xml:id="echoid-s6907" xml:space="preserve">11755705. </s>
            <s xml:id="echoid-s6908" xml:space="preserve">14142136. </s>
            <s xml:id="echoid-s6909" xml:space="preserve">17320508. </s>
            <s xml:id="echoid-s6910" xml:space="preserve">qualium 20000000. </s>
            <s xml:id="echoid-s6911" xml:space="preserve">tota diame-
              <lb/>
            ter ſtatuitur. </s>
            <s xml:id="echoid-s6912" xml:space="preserve">Ex chordis autem 6180340. </s>
            <s xml:id="echoid-s6913" xml:space="preserve">11755705. </s>
            <s xml:id="echoid-s6914" xml:space="preserve">arcuum grad. </s>
            <s xml:id="echoid-s6915" xml:space="preserve">36. </s>
            <s xml:id="echoid-s6916" xml:space="preserve">& </s>
            <s xml:id="echoid-s6917" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s6918" xml:space="preserve">
              <lb/>
            72. </s>
            <s xml:id="echoid-s6919" xml:space="preserve">inuenientur chordæ arcuum reliquorum in ſemicirculo, vt grad. </s>
            <s xml:id="echoid-s6920" xml:space="preserve">144. </s>
            <s xml:id="echoid-s6921" xml:space="preserve">& </s>
            <s xml:id="echoid-s6922" xml:space="preserve">
              <lb/>
            grad. </s>
            <s xml:id="echoid-s6923" xml:space="preserve">108. </s>
            <s xml:id="echoid-s6924" xml:space="preserve">partium 19021130. </s>
            <s xml:id="echoid-s6925" xml:space="preserve">16180340. </s>
            <s xml:id="echoid-s6926" xml:space="preserve">vt in propoſ. </s>
            <s xml:id="echoid-s6927" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6928" xml:space="preserve">oſtendimus.</s>
            <s xml:id="echoid-s6929" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6930" xml:space="preserve">DEINDE per propoſ. </s>
            <s xml:id="echoid-s6931" xml:space="preserve">13. </s>
            <s xml:id="echoid-s6932" xml:space="preserve">eiusq́ corollarium reperientur chordæ om-
              <lb/>
            nium arcuum, qui differentiæ ſint duorum quorumlibet arcuum, quorũ chor-
              <lb/>
            dæ ſint notæ. </s>
            <s xml:id="echoid-s6933" xml:space="preserve">Vt ex chorda arcus grad. </s>
            <s xml:id="echoid-s6934" xml:space="preserve">120. </s>
            <s xml:id="echoid-s6935" xml:space="preserve">& </s>
            <s xml:id="echoid-s6936" xml:space="preserve">ex chorda arcus grad. </s>
            <s xml:id="echoid-s6937" xml:space="preserve">36. </s>
            <s xml:id="echoid-s6938" xml:space="preserve">inue-
              <lb/>
            niemus chordam arcus grad. </s>
            <s xml:id="echoid-s6939" xml:space="preserve">84. </s>
            <s xml:id="echoid-s6940" xml:space="preserve">qui illorum differentia eſt. </s>
            <s xml:id="echoid-s6941" xml:space="preserve">Item ex </s>
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