Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div822" type="section" level="1" n="280">
          <pb o="289" file="319" n="319" rhead="LIBER SEXTVS."/>
          <p>
            <s xml:id="echoid-s13778" xml:space="preserve">
              <emph style="sc">Demonstro</emph>
            vtrumque hunc modum hac ratione. </s>
            <s xml:id="echoid-s13779" xml:space="preserve">Quando numerator
              <lb/>
            in quadratum denominatoris ducitur, erit producti numeriradix cubica vnus
              <lb/>
            duorum mediorum proportionalium inter numeratorem ac denominatorem
              <lb/>
            collo candus iuxta denominatorem, vt conſtat ex iis, quæ propoſ. </s>
            <s xml:id="echoid-s13780" xml:space="preserve">18. </s>
            <s xml:id="echoid-s13781" xml:space="preserve">huius lib.
              <lb/>
            </s>
            <s xml:id="echoid-s13782" xml:space="preserve">demonſtrata ſunt. </s>
            <s xml:id="echoid-s13783" xml:space="preserve"> Erit igitur proportio numeratoris ad denominatorem
              <note symbol="a" position="right" xlink:label="note-319-01" xlink:href="note-319-01a" xml:space="preserve">10. defiu.
                <lb/>
              quinti.</note>
            plicata proportionis radicis cubicæ inuentæ ad denominatorem: </s>
            <s xml:id="echoid-s13784" xml:space="preserve"> Eſt autem eadem proportio numeratoris ad denominatorem, tanquam cubi ad cubum,
              <lb/>
              <note symbol="b" position="right" xlink:label="note-319-02" xlink:href="note-319-02a" xml:space="preserve">12. octaui.</note>
            triplicata quoque proportionis radicis cubicæ numeratoris ad radicem cubi-
              <lb/>
            cam denominatoris. </s>
            <s xml:id="echoid-s13785" xml:space="preserve">Igitur erit radix cubica inuenta ad denominatorem, vt
              <lb/>
            radix cubica numeratoris ad radicem cubicam denominatoris: </s>
            <s xml:id="echoid-s13786" xml:space="preserve"> ac
              <note symbol="c" position="right" xlink:label="note-319-03" xlink:href="note-319-03a" xml:space="preserve">7. minutia-
                <lb/>
              rum ad finem
                <lb/>
              lib. 9.</note>
            minutia, cuius numerator radix cubica inuenta, ac denominator ipſe denomi-
              <lb/>
            nator, æqualis erit minutiæ, cuius numerator radix cubica numeratoris, ac
              <lb/>
            denominator radix cubica denominatoris. </s>
            <s xml:id="echoid-s13787" xml:space="preserve">Quam ob rem ſicut hæc minu-
              <lb/>
            tia eſt radix cubica fractionis propoſitæ, ita quoque illa eritradix cubica eiuſ-
              <lb/>
            dem fractionis. </s>
            <s xml:id="echoid-s13788" xml:space="preserve">Diuiſa ergo radice illa cubica inuenta per denominatorem
              <lb/>
            fractionis propoſitæ (quæ diuiſio fit, quia illa radix cubica inuenta eſt fractio,
              <lb/>
            ac proinde, vt cognoſcatur val or minutiæ, cuius numerator radix illa inuen-
              <lb/>
            ta, ac denominator ipſemet denominator fractionis propoſitæ, diuidendus eſt
              <lb/>
            numerator huius minutiæ per eius denominatorem: </s>
            <s xml:id="echoid-s13789" xml:space="preserve">alio quin ſi illa radix cu-
              <lb/>
            bica inuenta foret numerus integer, diuiſio facienda non eſſet) producetur ra-
              <lb/>
            dix cubica fractionis propoſitæ: </s>
            <s xml:id="echoid-s13790" xml:space="preserve">quemadmodum ex diuiſione radicis cubicæ
              <lb/>
            numeratoris per radicem cubicam denominatoris procreatur radix cubica fra-
              <lb/>
            ctionis propoſitæ: </s>
            <s xml:id="echoid-s13791" xml:space="preserve">quippe cum minutia nil aliud ſit, niſi numerator per deno-
              <lb/>
            minatorem diuiſus.</s>
            <s xml:id="echoid-s13792" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13793" xml:space="preserve">
              <emph style="sc">Qvando</emph>
            verò denominator in quadratum numeratoris ducitur, erit
              <lb/>
            numeri producti radix cubica vnus duorum mediorum proportionalium in-
              <lb/>
            ter numeratorem, ac denominatorem, collocandus iuxta numeratorem, vt
              <lb/>
            ex demonſtratis propoſ. </s>
            <s xml:id="echoid-s13794" xml:space="preserve">18. </s>
            <s xml:id="echoid-s13795" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s13796" xml:space="preserve">manifeſtum eſt. </s>
            <s xml:id="echoid-s13797" xml:space="preserve">Ergo rurſus erit propor-
              <lb/>
            tio numeratoris ad denominatorem triplicata proportionis numeratoris ad il-
              <lb/>
            lam radicem cubicam inuentam, quemadmodum & </s>
            <s xml:id="echoid-s13798" xml:space="preserve">proportionis radicis cu-
              <lb/>
            bicæ numeratoris ad radicem cubicam denominatoris, eadem illa proportio
              <lb/>
            numeratoris ad denominatorem, tanquam cubi ad cubum, triplicata eſt: </s>
            <s xml:id="echoid-s13799" xml:space="preserve">Ac
              <lb/>
            propterea, vt ſupra, minutia, cuius numerator ipſemet numerator fractio-
              <lb/>
            nis propoſitæ, denominator verò radix illa cubica inuenta, æqualis erit mi-
              <lb/>
            nutiæ, cuius numerator, radix cubica numeratoris fractionis propoſitæ, & </s>
            <s xml:id="echoid-s13800" xml:space="preserve">de-
              <lb/>
            nominator radix cubica denominatoris eiuſdem fractionis. </s>
            <s xml:id="echoid-s13801" xml:space="preserve">Quocirca diuiſo
              <lb/>
            numeratore per illam inuentam radicem cubicam, prodibit radix cubica fra-
              <lb/>
            ctionis propoſitæ.</s>
            <s xml:id="echoid-s13802" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div826" type="section" level="1" n="281">
          <head xml:id="echoid-head307" xml:space="preserve">PROBL. 17. PROPOS. 22.</head>
          <p>
            <s xml:id="echoid-s13803" xml:space="preserve">RADICEM quadratam & </s>
            <s xml:id="echoid-s13804" xml:space="preserve">cubicam in numeris non quadratis, & </s>
            <s xml:id="echoid-s13805" xml:space="preserve">
              <lb/>
            non cubicis per lineas Geometrice inuenire.</s>
            <s xml:id="echoid-s13806" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13807" xml:space="preserve">
              <emph style="sc">Sit</emph>
            datus numerus 10. </s>
            <s xml:id="echoid-s13808" xml:space="preserve">repræſentans quadratum 10. </s>
            <s xml:id="echoid-s13809" xml:space="preserve">palmorum, vel </s>
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