Clavius, Christoph, Geometria practica

Table of figures

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          <p>
            <s xml:id="echoid-s9831" xml:space="preserve">
              <pb o="219" file="249" n="249" rhead="LIBER QVINTVS."/>
            vt B, ad E, ita D, ad F; </s>
            <s xml:id="echoid-s9832" xml:space="preserve">quod vtrobique poſita ſit eadem proportio multiplex: </s>
            <s xml:id="echoid-s9833" xml:space="preserve">e-
              <lb/>
            rit ex æquo, vt A ad E, ita C, ad F. </s>
            <s xml:id="echoid-s9834" xml:space="preserve">quod eſt propoſitum.</s>
            <s xml:id="echoid-s9835" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9836" xml:space="preserve">
              <emph style="sc">Idem</emph>
            ſequitur ſi A, & </s>
            <s xml:id="echoid-s9837" xml:space="preserve">C, ſintipſarum B, D, eædem partes plures non facien-
              <lb/>
            tes vnam: </s>
            <s xml:id="echoid-s9838" xml:space="preserve">Item, ſi E, & </s>
            <s xml:id="echoid-s9839" xml:space="preserve">F, earundem B, D, ſint eædem partes plures non facien-
              <lb/>
            tes vnam, vt {2/3}. </s>
            <s xml:id="echoid-s9840" xml:space="preserve">vel {3/5}. </s>
            <s xml:id="echoid-s9841" xml:space="preserve">&</s>
            <s xml:id="echoid-s9842" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9843" xml:space="preserve">Nam ſi verbi gratia A, C, ſint {3/4}. </s>
            <s xml:id="echoid-s9844" xml:space="preserve">ipſarum B, D, erit
              <lb/>
            {1/4}. </s>
            <s xml:id="echoid-s9845" xml:space="preserve">ipſius B, ad B, vt {1/4}. </s>
            <s xml:id="echoid-s9846" xml:space="preserve">ipſius D, ad D. </s>
            <s xml:id="echoid-s9847" xml:space="preserve"> Igitur erunt quoq;</s>
            <s xml:id="echoid-s9848" xml:space="preserve">, vt {3/4}. </s>
            <s xml:id="echoid-s9849" xml:space="preserve">ipſius B,
              <note symbol="a" position="right" xlink:label="note-249-01" xlink:href="note-249-01a" xml:space="preserve">ſchol. 22.
                <lb/>
              quinti.</note>
            eſt, ipſa A, ad B, ita {3/4}. </s>
            <s xml:id="echoid-s9850" xml:space="preserve">ipſius D, hoc eſt, ipſa C, ad D. </s>
            <s xml:id="echoid-s9851" xml:space="preserve">Rurſus ſi verbi gratia E, F,
              <lb/>
            ſint {2/3}. </s>
            <s xml:id="echoid-s9852" xml:space="preserve">ipſarum B, D, erit, vt B, ad {1/3}. </s>
            <s xml:id="echoid-s9853" xml:space="preserve">eiuſdem B, ita D, ad {1/3}. </s>
            <s xml:id="echoid-s9854" xml:space="preserve">eiuſdem D. </s>
            <s xml:id="echoid-s9855" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-249-02" xlink:href="note-249-02a" xml:space="preserve">ſchol. 22.
                <lb/>
              quinti.</note>
            inde vt B, ad {2/3}. </s>
            <s xml:id="echoid-s9856" xml:space="preserve">id eſt, ad E, ita D, ad {2/3}. </s>
            <s xml:id="echoid-s9857" xml:space="preserve">id eſt, ad F. </s>
            <s xml:id="echoid-s9858" xml:space="preserve">Quare, vt prius, erit ex æquo,
              <lb/>
            vt A, ad E, ita C. </s>
            <s xml:id="echoid-s9859" xml:space="preserve">ad F.</s>
            <s xml:id="echoid-s9860" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div603" type="section" level="1" n="211">
          <head xml:id="echoid-head228" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s9861" xml:space="preserve">
              <emph style="sc">Seqvitvr</emph>
            hinc, ita eſſe {1/4}. </s>
            <s xml:id="echoid-s9862" xml:space="preserve">cuiuſuis magnitudinis ad {1/3}. </s>
            <s xml:id="echoid-s9863" xml:space="preserve">eiuſdem, vt eſt {1/2}.
              <lb/>
            </s>
            <s xml:id="echoid-s9864" xml:space="preserve">cuiuſuis alteri
              <emph style="sub">9</emph>
            magnitudinis ad {2/3}. </s>
            <s xml:id="echoid-s9865" xml:space="preserve">eiuſdẽ. </s>
            <s xml:id="echoid-s9866" xml:space="preserve">Quoniã. </s>
            <s xml:id="echoid-s9867" xml:space="preserve">n. </s>
            <s xml:id="echoid-s9868" xml:space="preserve">vt oſtendim
              <emph style="sub">9</emph>
            , ita eſt {1/4}. </s>
            <s xml:id="echoid-s9869" xml:space="preserve">prio-
              <lb/>
            ris magnitudinis ad {1/3}. </s>
            <s xml:id="echoid-s9870" xml:space="preserve">eiuſdem, vt {1/4}. </s>
            <s xml:id="echoid-s9871" xml:space="preserve">poſterioris ad {1/3}. </s>
            <s xml:id="echoid-s9872" xml:space="preserve">eiuſdem. </s>
            <s xml:id="echoid-s9873" xml:space="preserve"> Vtautem {1/4}.</s>
            <s xml:id="echoid-s9874" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-249-03" xlink:href="note-249-03a" xml:space="preserve">1. quiuti.</note>
            poſterioris ad {1/3}. </s>
            <s xml:id="echoid-s9875" xml:space="preserve">ita ſunt {2/4}. </s>
            <s xml:id="echoid-s9876" xml:space="preserve">ad {2/3}. </s>
            <s xml:id="echoid-s9877" xml:space="preserve">hoc eſt, {1/2}. </s>
            <s xml:id="echoid-s9878" xml:space="preserve">ad {2/3}. </s>
            <s xml:id="echoid-s9879" xml:space="preserve">Igitur erit vt {1/4}. </s>
            <s xml:id="echoid-s9880" xml:space="preserve">prioris
              <lb/>
            magnitudinis ad {1/3}. </s>
            <s xml:id="echoid-s9881" xml:space="preserve">eiuſdem, ita {1/2}. </s>
            <s xml:id="echoid-s9882" xml:space="preserve">poſterioris magnitudinis ad {2/3}. </s>
            <s xml:id="echoid-s9883" xml:space="preserve">eiuſdem.</s>
            <s xml:id="echoid-s9884" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div605" type="section" level="1" n="212">
          <head xml:id="echoid-head229" xml:space="preserve">PROPOSITIO II.</head>
          <p>
            <s xml:id="echoid-s9885" xml:space="preserve">RECTANGVLVM ſub diametro, & </s>
            <s xml:id="echoid-s9886" xml:space="preserve">circumferentia maximi circuli
              <lb/>
            in ſphæra comprehenſum, quadruplum eſt circuli maximi, & </s>
            <s xml:id="echoid-s9887" xml:space="preserve">ſuper-
              <lb/>
            ficiei conuexæ eiuſdem ſphęræ ęquale.</s>
            <s xml:id="echoid-s9888" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9889" xml:space="preserve">
              <emph style="sc">Sit</emph>
            rectangulum AB, comprehenſum ſub diametro AC, & </s>
            <s xml:id="echoid-s9890" xml:space="preserve">circumferentia
              <lb/>
            CB, maximi in ſphæra circuli. </s>
            <s xml:id="echoid-s9891" xml:space="preserve">Dico rectangulum AB,
              <lb/>
              <figure xlink:label="fig-249-01" xlink:href="fig-249-01a" number="162">
                <image file="249-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/249-01"/>
              </figure>
            quadruplum eſſe circuli maximi in ſphæra, & </s>
            <s xml:id="echoid-s9892" xml:space="preserve">ſuperficiei
              <lb/>
            conuexæ eiuſdem ſphæræ ęquale. </s>
            <s xml:id="echoid-s9893" xml:space="preserve">Sectis enim omnibus
              <lb/>
            lateribus bifariam in E, F, G, H, iunctiſquerectis EG, FH,
              <lb/>
            ſecantibus ſeſe in I, diuiſum erit totum rectangulum in
              <lb/>
            quatuor æqualia A I, C I, B I, D I, quodrectæ E G, F
              <note symbol="d" position="right" xlink:label="note-249-04" xlink:href="note-249-04a" xml:space="preserve">33. primi.</note>
            rectis A D, A C, parallelæ ſint. </s>
            <s xml:id="echoid-s9894" xml:space="preserve">Ac proinderectangulum A B, rectanguli C I,
              <lb/>
            quadruplum erit. </s>
            <s xml:id="echoid-s9895" xml:space="preserve">Eſt autem rectangulum C I, contentum ſub C E, ſemidia-
              <lb/>
            metro, & </s>
            <s xml:id="echoid-s9896" xml:space="preserve">ſemicircumferentia C F, circulo maximo, cuius nimirum diameter
              <lb/>
            A C, ęquale, vt lib. </s>
            <s xml:id="echoid-s9897" xml:space="preserve">4. </s>
            <s xml:id="echoid-s9898" xml:space="preserve">capit. </s>
            <s xml:id="echoid-s9899" xml:space="preserve">7. </s>
            <s xml:id="echoid-s9900" xml:space="preserve">Nume. </s>
            <s xml:id="echoid-s9901" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9902" xml:space="preserve">demonſtratum eſt. </s>
            <s xml:id="echoid-s9903" xml:space="preserve">lgitur rectangu-
              <lb/>
            lum A B, circulimaximi quadruplum eſt. </s>
            <s xml:id="echoid-s9904" xml:space="preserve">Et quia eiuſdem circuli maximi qua-
              <lb/>
            drupla eſt ſuperficies conuexa ſphærę, per propoſ. </s>
            <s xml:id="echoid-s9905" xml:space="preserve">31. </s>
            <s xml:id="echoid-s9906" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s9907" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9908" xml:space="preserve">Archimedis de ſphę-
              <lb/>
            ra, & </s>
            <s xml:id="echoid-s9909" xml:space="preserve">Cylindro: </s>
            <s xml:id="echoid-s9910" xml:space="preserve"> ęquale erit rectangulum A B, conuexæ ſuperficiei, quod
              <note symbol="e" position="right" xlink:label="note-249-05" xlink:href="note-249-05a" xml:space="preserve">9. quinti.</note>
            rat demonſtrandum.</s>
            <s xml:id="echoid-s9911" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div607" type="section" level="1" n="213">
          <head xml:id="echoid-head230" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s9912" xml:space="preserve">EX demonſtratione liquet, rectangulum ſub diametro cuiuſuis circuli, (et-
              <lb/>
            iamſi non ſit maximus in ſphęra,) & </s>
            <s xml:id="echoid-s9913" xml:space="preserve">circumferentia eiuſdem, quadruplum eſſe
              <lb/>
            ipſius circuli. </s>
            <s xml:id="echoid-s9914" xml:space="preserve">Eadem enim ſemper demonſtratio adhibebitur.</s>
            <s xml:id="echoid-s9915" xml:space="preserve"/>
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