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

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        <div xml:id="echoid-div348" type="section" level="1" n="153">
          <p>
            <s xml:id="echoid-s4146" xml:space="preserve">
              <pb o="111" file="123" n="123" rhead=""/>
            coll. </s>
            <s xml:id="echoid-s4147" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4148" xml:space="preserve">15. </s>
            <s xml:id="echoid-s4149" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4150" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4151" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4152" xml:space="preserve">erit minus ſegmentum DF, latus Decagoni in eo-
              <lb/>
            dem circulo, vt ad propoſ. </s>
            <s xml:id="echoid-s4153" xml:space="preserve">9. </s>
            <s xml:id="echoid-s4154" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4155" xml:space="preserve">13. </s>
            <s xml:id="echoid-s4156" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4157" xml:space="preserve">demonſtrauimus. </s>
            <s xml:id="echoid-s4158" xml:space="preserve">Rurſus quoniam
              <lb/>
            quadrato lateris Hexagoni BD, vna cum quadrato lateris Decagoni DF,
              <lb/>
              <note position="right" xlink:label="note-123-01" xlink:href="note-123-01a" xml:space="preserve">10. tertij-
                <lb/>
              dec.</note>
            æquale eſt quadratum lateris Pentagoni in eodem circulo: </s>
            <s xml:id="echoid-s4159" xml:space="preserve">Eſt autem eiſdem
              <lb/>
            quadratis rectarum BD, DF, æquale quadratum rectæ BF; </s>
            <s xml:id="echoid-s4160" xml:space="preserve">erit quadratum
              <lb/>
              <note position="right" xlink:label="note-123-02" xlink:href="note-123-02a" xml:space="preserve">47. primi.</note>
            lateris Pentagoni æquale quadrato rectæ BF; </s>
            <s xml:id="echoid-s4161" xml:space="preserve">ac propterea recta BF, lateri
              <lb/>
            Pentagoni æqualis. </s>
            <s xml:id="echoid-s4162" xml:space="preserve">Latera igitur Decagoni, & </s>
            <s xml:id="echoid-s4163" xml:space="preserve">Pentagoni æquilateri in vno
              <lb/>
            @odemq́ue circulo inueſtigauimus. </s>
            <s xml:id="echoid-s4164" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s4165" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div351" type="section" level="1" n="154">
          <head xml:id="echoid-head180" xml:space="preserve">PROBL. 2. PROPOS. 3.</head>
          <note position="right" xml:space="preserve">Ex ſinu re-
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          ctocuiuſuis
            <lb/>
          arcus quo
            <lb/>
          pacto ſinus
            <lb/>
          com plemé
            <lb/>
          ti eiuſdem
            <lb/>
          arcus, & ex
            <lb/>
          chorda cu-
            <lb/>
          iuſuis ar -
            <lb/>
          cus qua ra-
            <lb/>
          tione chor-
            <lb/>
          da reliqui
            <lb/>
          arcus ſemi-
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          circuli co-
            <lb/>
          gnoſcatur.
            <lb/>
          47. primi.</note>
          <p>
            <s xml:id="echoid-s4166" xml:space="preserve">EX ſinu recto cuiuſuis arcus quadrante mi-
              <lb/>
            noris cognito, ſinum complementi eiuſdem ar-
              <lb/>
            cus; </s>
            <s xml:id="echoid-s4167" xml:space="preserve">lté ex chorda cuiuſuis arcus ſemicirculo mino
              <lb/>
            ris, chordam rehqui arcus ſemicircuh cognoſcere.</s>
            <s xml:id="echoid-s4168" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4169" xml:space="preserve">SIT primo cognitus ſinus rectus DE, arcus BD, cuius arcus complemen
              <lb/>
            ti ſinus ſit DF, quem cognoſcere debemus. </s>
            <s xml:id="echoid-s4170" xml:space="preserve">Ducta recta DA, erit quadratum
              <lb/>
            rectæ DA, æquale quadratis rectarum DE, EA. </s>
            <s xml:id="echoid-s4171" xml:space="preserve">Si igitur ex quadrato ſinus
              <lb/>
            totius DA, noti (Ponitur enim ſinus totus particularum certo numero com-
              <lb/>
            prehenſarum) detrahatur
              <lb/>
              <figure xlink:label="fig-123-01" xlink:href="fig-123-01a" number="120">
                <image file="123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/123-01"/>
              </figure>
            quadratũ ſinus recti DE,
              <lb/>
            cogniti in partibus ſinus
              <lb/>
            totius DA, relinquetur
              <lb/>
            quadratum rectæ EA, no
              <lb/>
            tum; </s>
            <s xml:id="echoid-s4172" xml:space="preserve">ac proinde per radi-
              <lb/>
            cem quadratam recta EA,
              <lb/>
            in eiſdem partibus nota
              <lb/>
            erit. </s>
            <s xml:id="echoid-s4173" xml:space="preserve">Cum ergo recta EA,
              <lb/>
            æqualis ſit ſinui comple-
              <lb/>
            menti arcus BD, hoc eſt,
              <lb/>
              <note position="right" xlink:label="note-123-04" xlink:href="note-123-04a" xml:space="preserve">34. primi.</note>
            rectæ DF, cognitus erit
              <lb/>
            DF, ſinus complementi arcus BD, cuius ſinus rectus DE, notus eſt poſitus.</s>
            <s xml:id="echoid-s4174" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4175" xml:space="preserve">SIT deinde cognita chorda AB, arcus AB, & </s>
            <s xml:id="echoid-s4176" xml:space="preserve">chorda BC, ſubtendens re-
              <lb/>
            liquum arcum BC, ſemicirculi, quam iubemur inueſtigare. </s>
            <s xml:id="echoid-s4177" xml:space="preserve">Quoniam angulus
              <lb/>
              <note position="right" xlink:label="note-123-05" xlink:href="note-123-05a" xml:space="preserve">31. tertij.</note>
            B, rectus eſt in ſemicirculo, erit quadratum diametri AC, æquale quadratis
              <lb/>
              <note position="right" xlink:label="note-123-06" xlink:href="note-123-06a" xml:space="preserve">47. primi.</note>
            chordarum AB, BC. </s>
            <s xml:id="echoid-s4178" xml:space="preserve">Si igitur ex quadrato diametri AC, notæ (Ponitur enim
              <lb/>
            diameter diuiſa in particulas certo numero comprehenſas) dematur quadra-
              <lb/>
            tum chordæ AB, notæ in partibus diametri AC, notum relinquetur quadra-
              <lb/>
            tum chordæ BC; </s>
            <s xml:id="echoid-s4179" xml:space="preserve">ac proinde per radicem quadratam chorda BC, in eiſdem
              <lb/>
            partibus nota efficietur. </s>
            <s xml:id="echoid-s4180" xml:space="preserve">Ex ſinu igitur recto cuiuſuis arcus, &</s>
            <s xml:id="echoid-s4181" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4182" xml:space="preserve">cognouimus.
              <lb/>
            </s>
            <s xml:id="echoid-s4183" xml:space="preserve">Quod ſaciendum erat.</s>
            <s xml:id="echoid-s4184" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div354" type="section" level="1" n="155">
          <head xml:id="echoid-head181" xml:space="preserve">COROLLARIVM.</head>
          <note position="right" xml:space="preserve">Sinus vet-
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          ſus cogno-
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          ſcitur ex co
            <lb/>
          gnito ſinu
            <lb/>
          recto,.</note>
          <p>
            <s xml:id="echoid-s4185" xml:space="preserve">HINC efficitur, ſinum verſum cuiuſuis arcus cognoſci quoque ex cognito ſinu recto.
              <lb/>
            </s>
            <s xml:id="echoid-s4186" xml:space="preserve">Quoniam enim ex ſinu recto DE, cognoſcitur ſinus complementi DP, hoc eſt, AE; </s>
            <s xml:id="echoid-s4187" xml:space="preserve">ſi </s>
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