Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div981" type="section" level="1" n="350">
          <pb o="343" file="371" n="371" rhead="LIBER OCTAVVS."/>
        </div>
        <div xml:id="echoid-div983" type="section" level="1" n="351">
          <head xml:id="echoid-head378" xml:space="preserve">THEOR. 5. PROPOS. 10.</head>
          <p>
            <s xml:id="echoid-s16031" xml:space="preserve">PROPOSITIS duabus minutiis inæqualibus; </s>
            <s xml:id="echoid-s16032" xml:space="preserve">minutia, cuius nume-
              <lb/>
            rator ex illarum numeratoribus, denominator autem ex denomina-
              <lb/>
            toribus conflatur, maior quidem eſt minore, minor vero maiore.</s>
            <s xml:id="echoid-s16033" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16034" xml:space="preserve">
              <emph style="sc">Sint</emph>
            duæ minutiæ inæquales, maior {3/5}. </s>
            <s xml:id="echoid-s16035" xml:space="preserve">& </s>
            <s xml:id="echoid-s16036" xml:space="preserve">minor {4/7}. </s>
            <s xml:id="echoid-s16037" xml:space="preserve">Iungantur tam nume-
              <lb/>
            tatores, quam denominatores, vt fiat minutia {7/12}. </s>
            <s xml:id="echoid-s16038" xml:space="preserve">Dico hanc maiorem eſſe,
              <lb/>
            quam {4/7}. </s>
            <s xml:id="echoid-s16039" xml:space="preserve">& </s>
            <s xml:id="echoid-s16040" xml:space="preserve">minorem quam {3/5}. </s>
            <s xml:id="echoid-s16041" xml:space="preserve">Quoniam enim maior eſt minutia {3/5}. </s>
            <s xml:id="echoid-s16042" xml:space="preserve">quam {4/7}.
              <lb/>
            </s>
            <s xml:id="echoid-s16043" xml:space="preserve">erit per propoſ. </s>
            <s xml:id="echoid-s16044" xml:space="preserve">8. </s>
            <s xml:id="echoid-s16045" xml:space="preserve">Minutiarum lib. </s>
            <s xml:id="echoid-s16046" xml:space="preserve">9. </s>
            <s xml:id="echoid-s16047" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s16048" xml:space="preserve">maior proportio 3. </s>
            <s xml:id="echoid-s16049" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s16050" xml:space="preserve">quam 4. </s>
            <s xml:id="echoid-s16051" xml:space="preserve">ad
              <lb/>
            7. </s>
            <s xml:id="echoid-s16052" xml:space="preserve"> Et permutando, maior 3. </s>
            <s xml:id="echoid-s16053" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s16054" xml:space="preserve">quam 5. </s>
            <s xml:id="echoid-s16055" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s16056" xml:space="preserve"> Igitur &</s>
            <s xml:id="echoid-s16057" xml:space="preserve">
              <note position="right" xlink:label="note-371-01" xlink:href="note-371-01a" xml:space="preserve">
                <lb/>
              {3/5}. # {4/7}. # {7/12}.
                <lb/>
              </note>
              <note symbol="a" position="right" xlink:label="note-371-02" xlink:href="note-371-02a" xml:space="preserve">27 quinti.</note>
            componendo, maior 3. </s>
            <s xml:id="echoid-s16058" xml:space="preserve">4. </s>
            <s xml:id="echoid-s16059" xml:space="preserve">ſimul, hoc eſt, 7. </s>
            <s xml:id="echoid-s16060" xml:space="preserve">ad 4. </s>
            <s xml:id="echoid-s16061" xml:space="preserve">quam 5. </s>
            <s xml:id="echoid-s16062" xml:space="preserve">7.
              <lb/>
            </s>
            <s xml:id="echoid-s16063" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-371-03" xlink:href="note-371-03a" xml:space="preserve">28. quinti.</note>
            ſimul, id eſt, quam 12. </s>
            <s xml:id="echoid-s16064" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s16065" xml:space="preserve"> Et permutando, maior 7. </s>
            <s xml:id="echoid-s16066" xml:space="preserve">ad 12.</s>
            <s xml:id="echoid-s16067" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-371-04" xlink:href="note-371-04a" xml:space="preserve">27. quinti.</note>
            quam 4. </s>
            <s xml:id="echoid-s16068" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s16069" xml:space="preserve">Ac proinde per propoſ. </s>
            <s xml:id="echoid-s16070" xml:space="preserve">8. </s>
            <s xml:id="echoid-s16071" xml:space="preserve">Minutiarum libri 9. </s>
            <s xml:id="echoid-s16072" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s16073" xml:space="preserve">maior erit
              <lb/>
            minutia {7/12}. </s>
            <s xml:id="echoid-s16074" xml:space="preserve">quam {4/7}. </s>
            <s xml:id="echoid-s16075" xml:space="preserve">quod eſt primum.</s>
            <s xml:id="echoid-s16076" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16077" xml:space="preserve">
              <emph style="sc">Deinde</emph>
            quia minor eſt {4/7}. </s>
            <s xml:id="echoid-s16078" xml:space="preserve">quam {3/5}. </s>
            <s xml:id="echoid-s16079" xml:space="preserve">erit per propoſ. </s>
            <s xml:id="echoid-s16080" xml:space="preserve">8. </s>
            <s xml:id="echoid-s16081" xml:space="preserve">Minutiarum libri
              <lb/>
            9. </s>
            <s xml:id="echoid-s16082" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s16083" xml:space="preserve">minor proportio 4. </s>
            <s xml:id="echoid-s16084" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s16085" xml:space="preserve">quam 3. </s>
            <s xml:id="echoid-s16086" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s16087" xml:space="preserve"> & </s>
            <s xml:id="echoid-s16088" xml:space="preserve">permutando, minor 4. </s>
            <s xml:id="echoid-s16089" xml:space="preserve">
              <note symbol="d" position="right" xlink:label="note-371-05" xlink:href="note-371-05a" xml:space="preserve">27. quinti.</note>
            3. </s>
            <s xml:id="echoid-s16090" xml:space="preserve">quam 7. </s>
            <s xml:id="echoid-s16091" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s16092" xml:space="preserve"> Igitur & </s>
            <s xml:id="echoid-s16093" xml:space="preserve">componendo minor 4. </s>
            <s xml:id="echoid-s16094" xml:space="preserve">3. </s>
            <s xml:id="echoid-s16095" xml:space="preserve">ſimul, id eſt, 7. </s>
            <s xml:id="echoid-s16096" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s16097" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-371-06" xlink:href="note-371-06a" xml:space="preserve">28. quinti.</note>
            7. </s>
            <s xml:id="echoid-s16098" xml:space="preserve">5. </s>
            <s xml:id="echoid-s16099" xml:space="preserve">ſimul, hoc eſt. </s>
            <s xml:id="echoid-s16100" xml:space="preserve">quam 12. </s>
            <s xml:id="echoid-s16101" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s16102" xml:space="preserve"> Et permutando, minor 7. </s>
            <s xml:id="echoid-s16103" xml:space="preserve">ad 12. </s>
            <s xml:id="echoid-s16104" xml:space="preserve">quam 3. </s>
            <s xml:id="echoid-s16105" xml:space="preserve">
              <note symbol="f" position="right" xlink:label="note-371-07" xlink:href="note-371-07a" xml:space="preserve">27. quinti.</note>
            5. </s>
            <s xml:id="echoid-s16106" xml:space="preserve">Ac proinde per propoſ. </s>
            <s xml:id="echoid-s16107" xml:space="preserve">8. </s>
            <s xml:id="echoid-s16108" xml:space="preserve">Minutiarum lib. </s>
            <s xml:id="echoid-s16109" xml:space="preserve">9. </s>
            <s xml:id="echoid-s16110" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s16111" xml:space="preserve">minor erit minutia {7/12}. </s>
            <s xml:id="echoid-s16112" xml:space="preserve">quã
              <lb/>
            {3/5}. </s>
            <s xml:id="echoid-s16113" xml:space="preserve">quod eſt ſecundum.</s>
            <s xml:id="echoid-s16114" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div986" type="section" level="1" n="352">
          <head xml:id="echoid-head379" xml:space="preserve">THEOR. 6. PROPOS. 11.</head>
          <p>
            <s xml:id="echoid-s16115" xml:space="preserve">SI duonumeri inter ſe primi non ſint ambo quadrati aut cubi; </s>
            <s xml:id="echoid-s16116" xml:space="preserve">neque
              <lb/>
            eorum æquè multiplices vlli, quadrati erunt, aut cubi. </s>
            <s xml:id="echoid-s16117" xml:space="preserve">Et ſi eorum æ-
              <lb/>
            què multiplices aliqui ſint ambo quadrati, aut cubi, etiam ipſi erunt
              <lb/>
            quadrati aut cubi.</s>
            <s xml:id="echoid-s16118" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16119" xml:space="preserve">
              <emph style="sc">Sint</emph>
            enim A, B, numeri inter ſe primi, & </s>
            <s xml:id="echoid-s16120" xml:space="preserve">non ambo quadrati, vel cubi, quã-
              <lb/>
            uis vnus eorum qua dratus ſit, vel cubus. </s>
            <s xml:id="echoid-s16121" xml:space="preserve">ſintque eorum æ-
              <lb/>
              <note position="right" xlink:label="note-371-08" xlink:href="note-371-08a" xml:space="preserve">
                <lb/>
              A # B
                <lb/>
              4. # 11.
                <lb/>
              C # D
                <lb/>
              12. # 33.
                <lb/>
              </note>
            què multiplices C, D. </s>
            <s xml:id="echoid-s16122" xml:space="preserve">Dico neq; </s>
            <s xml:id="echoid-s16123" xml:space="preserve">hos eſſe ambos quadra-
              <lb/>
            tos, aut cubos. </s>
            <s xml:id="echoid-s16124" xml:space="preserve">Sint enim, ſi fieri poteſt, ambo quadrati,
              <lb/>
            vel cubi. </s>
            <s xml:id="echoid-s16125" xml:space="preserve">Et quoniam idem numerus multiplicans A, & </s>
            <s xml:id="echoid-s16126" xml:space="preserve">B,
              <lb/>
            fecit C, & </s>
            <s xml:id="echoid-s16127" xml:space="preserve">D, quod hi illorum ſint æque multiplices: </s>
            <s xml:id="echoid-s16128" xml:space="preserve">
              <note symbol="g" position="right" xlink:label="note-371-09" xlink:href="note-371-09a" xml:space="preserve">17. ſeptimi.</note>
            A, ad B, vt C, ad D. </s>
            <s xml:id="echoid-s16129" xml:space="preserve"> Cadit autem inter C, & </s>
            <s xml:id="echoid-s16130" xml:space="preserve">D, vnus
              <note symbol="h" position="right" xlink:label="note-371-10" xlink:href="note-371-10a" xml:space="preserve">11. & 12.
                <lb/>
              octaui.</note>
            dius proportionalis, aut duo. </s>
            <s xml:id="echoid-s16131" xml:space="preserve"> Igitur & </s>
            <s xml:id="echoid-s16132" xml:space="preserve">inter A, B, vnus
              <note symbol="i" position="right" xlink:label="note-371-11" xlink:href="note-371-11a" xml:space="preserve">8. octaui.</note>
            det medius proportionalis, aut duo. </s>
            <s xml:id="echoid-s16133" xml:space="preserve">Cum ergo extremi A,
              <lb/>
            B, ponantur inter ſe primi; </s>
            <s xml:id="echoid-s16134" xml:space="preserve"> erunt omnes tres, vel
              <note symbol="k" position="right" xlink:label="note-371-12" xlink:href="note-371-12a" xml:space="preserve">1. octaui.</note>
            proportionales, minimi in ſua proportione: </s>
            <s xml:id="echoid-s16135" xml:space="preserve"> Ac
              <note symbol="l" position="right" xlink:label="note-371-13" xlink:href="note-371-13a" xml:space="preserve">coroll. 2.
                <lb/>
              octaui.</note>
            A, B, ambo qua drati erunt, vel cubi. </s>
            <s xml:id="echoid-s16136" xml:space="preserve">quod eſt contra hy-
              <lb/>
            potheſim. </s>
            <s xml:id="echoid-s16137" xml:space="preserve">Non ergo C, D, ambo qua drati ſunt, aut cubi. </s>
            <s xml:id="echoid-s16138" xml:space="preserve">quod erat oſtenden-
              <lb/>
            dum.</s>
            <s xml:id="echoid-s16139" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s16140" xml:space="preserve">
              <emph style="sc">Sed</emph>
            ſintiam C, D, ipſorum A, B, inter ſe primorum æquè multiplices, & </s>
            <s xml:id="echoid-s16141" xml:space="preserve">am-
              <lb/>
            bo quadrati, vel cubi. </s>
            <s xml:id="echoid-s16142" xml:space="preserve">Dico etiam A, B, ambos eſſe quadratos, vel cubos. </s>
            <s xml:id="echoid-s16143" xml:space="preserve">Si. </s>
            <s xml:id="echoid-s16144" xml:space="preserve">n.</s>
            <s xml:id="echoid-s16145" xml:space="preserve"/>
          </p>
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