Clavius, Christoph, Geometria practica

Table of handwritten notes

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        <div xml:id="echoid-div815" type="section" level="1" n="279">
          <pb o="286" file="316" n="316" rhead="GEOMETR. PRACT."/>
          <p>
            <s xml:id="echoid-s13530" xml:space="preserve">
              <emph style="sc">Qvando</emph>
            autem apponuntur 0000. </s>
            <s xml:id="echoid-s13531" xml:space="preserve">multiplicatur numerus propoſitus,
              <lb/>
            verbi gratia numerus 5. </s>
            <s xml:id="echoid-s13532" xml:space="preserve">per 10000. </s>
            <s xml:id="echoid-s13533" xml:space="preserve">id eſt, per quadratum radicis 100. </s>
            <s xml:id="echoid-s13534" xml:space="preserve">Et quia
              <lb/>
            quadrati 50000. </s>
            <s xml:id="echoid-s13535" xml:space="preserve">5. </s>
            <s xml:id="echoid-s13536" xml:space="preserve"> habent duplicatam proportionem ſuarum radicum. </s>
            <s xml:id="echoid-s13537" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-316-01" xlink:href="note-316-01a" xml:space="preserve">11. octaui.</note>
            autem 50000. </s>
            <s xml:id="echoid-s13538" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s13539" xml:space="preserve">vt 10000. </s>
            <s xml:id="echoid-s13540" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13541" xml:space="preserve">propterea quod| 5. </s>
            <s xml:id="echoid-s13542" xml:space="preserve">multiplicatus per 10000.
              <lb/>
            </s>
            <s xml:id="echoid-s13543" xml:space="preserve">fecit 50000. </s>
            <s xml:id="echoid-s13544" xml:space="preserve">Proportio autem 10000. </s>
            <s xml:id="echoid-s13545" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13546" xml:space="preserve">duplicata eſt proportionis 1000. </s>
            <s xml:id="echoid-s13547" xml:space="preserve">
              <lb/>
            ad 1. </s>
            <s xml:id="echoid-s13548" xml:space="preserve">vt in hoc exemplo patet; </s>
            <s xml:id="echoid-s13549" xml:space="preserve">erit proportio ra-
              <lb/>
              <note position="right" xlink:label="note-316-02" xlink:href="note-316-02a" xml:space="preserve">
                <lb/>
              1. # 100. # 10000.
                <lb/>
              # 5. # 50000.
                <lb/>
              </note>
            dicis numeri 50000. </s>
            <s xml:id="echoid-s13550" xml:space="preserve">nimirum 223. </s>
            <s xml:id="echoid-s13551" xml:space="preserve">ad radicem nu-
              <lb/>
            meri 5. </s>
            <s xml:id="echoid-s13552" xml:space="preserve">vt 100. </s>
            <s xml:id="echoid-s13553" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13554" xml:space="preserve">Quare ſi radix 223. </s>
            <s xml:id="echoid-s13555" xml:space="preserve">diuidatur
              <lb/>
            per 100. </s>
            <s xml:id="echoid-s13556" xml:space="preserve">procreabitur radix quadrata propinqua
              <lb/>
            2 {23/100}. </s>
            <s xml:id="echoid-s13557" xml:space="preserve">minor quam vera, at 2 {24/100}. </s>
            <s xml:id="echoid-s13558" xml:space="preserve">erit maior quam
              <lb/>
            vera. </s>
            <s xml:id="echoid-s13559" xml:space="preserve">Rectè ergo præcepimus, cum apponuntur 0000. </s>
            <s xml:id="echoid-s13560" xml:space="preserve">abiiciendas eſſe ex radi-
              <lb/>
            ce 223. </s>
            <s xml:id="echoid-s13561" xml:space="preserve">duas figuras, vt reliqua fiatradix 2 {23/100}.</s>
            <s xml:id="echoid-s13562" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13563" xml:space="preserve">
              <emph style="sc">Pari</emph>
            ratione, ſi apponantur 000000. </s>
            <s xml:id="echoid-s13564" xml:space="preserve">procreabitur radix propinqua in
              <lb/>
            milleſimis, at queita deinceps.</s>
            <s xml:id="echoid-s13565" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13566" xml:space="preserve">
              <emph style="sc">Rvrsvs</emph>
            quando pro radice cubica ad numerum propoſitum, vt ad 5. </s>
            <s xml:id="echoid-s13567" xml:space="preserve">ad-
              <lb/>
            duntur 000. </s>
            <s xml:id="echoid-s13568" xml:space="preserve">multiplicatur datus numerus per 1000. </s>
            <s xml:id="echoid-s13569" xml:space="preserve">id eſt, per cubum radicis
              <lb/>
            10. </s>
            <s xml:id="echoid-s13570" xml:space="preserve">Et quia cubi 5000. </s>
            <s xml:id="echoid-s13571" xml:space="preserve">5. </s>
            <s xml:id="echoid-s13572" xml:space="preserve"> habentproportionem ſuarum radicum triplicatam:</s>
            <s xml:id="echoid-s13573" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-316-03" xlink:href="note-316-03a" xml:space="preserve">12. octaui.</note>
            Eſt autem 5000. </s>
            <s xml:id="echoid-s13574" xml:space="preserve">ad 5. </s>
            <s xml:id="echoid-s13575" xml:space="preserve">vt 1000. </s>
            <s xml:id="echoid-s13576" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13577" xml:space="preserve">quod 1000. </s>
            <s xml:id="echoid-s13578" xml:space="preserve">mul-
              <lb/>
            tiplicans 5. </s>
            <s xml:id="echoid-s13579" xml:space="preserve">fecit 5000. </s>
            <s xml:id="echoid-s13580" xml:space="preserve">Proportio autem 1000. </s>
            <s xml:id="echoid-s13581" xml:space="preserve">ad 1.
              <lb/>
            </s>
            <s xml:id="echoid-s13582" xml:space="preserve">
              <note position="right" xlink:label="note-316-04" xlink:href="note-316-04a" xml:space="preserve">
                <lb/>
              1. # 10. # 100. # 1000.
                <lb/>
              # # 5. # 5000.
                <lb/>
              </note>
            triplicata eſt proportionis 10. </s>
            <s xml:id="echoid-s13583" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13584" xml:space="preserve">vt in hoc appoſito
              <lb/>
            exemplo apparet; </s>
            <s xml:id="echoid-s13585" xml:space="preserve">erit proportio radicis numeri 5000.
              <lb/>
            </s>
            <s xml:id="echoid-s13586" xml:space="preserve">nimirum 17. </s>
            <s xml:id="echoid-s13587" xml:space="preserve">ad radicem numeri 5. </s>
            <s xml:id="echoid-s13588" xml:space="preserve">vt 10. </s>
            <s xml:id="echoid-s13589" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13590" xml:space="preserve">Quo-
              <lb/>
            circa ſi radix 17. </s>
            <s xml:id="echoid-s13591" xml:space="preserve">diuidatur per 10. </s>
            <s xml:id="echoid-s13592" xml:space="preserve">fit radix cubica propinqua 1 {7/10}. </s>
            <s xml:id="echoid-s13593" xml:space="preserve">minor quam
              <lb/>
            vera, at 1 {8/10}. </s>
            <s xml:id="echoid-s13594" xml:space="preserve">erit maior quam vera. </s>
            <s xml:id="echoid-s13595" xml:space="preserve">Rectè ergo præcepimus, quando pro radice
              <lb/>
            cubica apponuntur 000. </s>
            <s xml:id="echoid-s13596" xml:space="preserve">abiiciendam eſſe ex radice inuenta 17. </s>
            <s xml:id="echoid-s13597" xml:space="preserve">vnam figuram,
              <lb/>
            vt reliqua fiatradix 1 {7/10}.</s>
            <s xml:id="echoid-s13598" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13599" xml:space="preserve">
              <emph style="sc">Simili</emph>
            modo ſi apponantur 000000. </s>
            <s xml:id="echoid-s13600" xml:space="preserve">inuenietur propinqua radix in cen-
              <lb/>
            teſimis: </s>
            <s xml:id="echoid-s13601" xml:space="preserve">Et ſi apponantur 000000000. </s>
            <s xml:id="echoid-s13602" xml:space="preserve">in milleſimis, &</s>
            <s xml:id="echoid-s13603" xml:space="preserve">c. </s>
            <s xml:id="echoid-s13604" xml:space="preserve">Nam ibi multipli-
              <lb/>
            catur numerus per 1000000. </s>
            <s xml:id="echoid-s13605" xml:space="preserve">id eſt, per cubum radicis 100. </s>
            <s xml:id="echoid-s13606" xml:space="preserve">hic verò per
              <lb/>
            1000000000. </s>
            <s xml:id="echoid-s13607" xml:space="preserve">nimirum per cubum radicis 1000. </s>
            <s xml:id="echoid-s13608" xml:space="preserve">Cætera eodem modo demon-
              <lb/>
            ſtrabuntur.</s>
            <s xml:id="echoid-s13609" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13610" xml:space="preserve">
              <emph style="sc">Neqve</emph>
            verò diuerſa ratio eſt in aliis radicibus. </s>
            <s xml:id="echoid-s13611" xml:space="preserve">Nam in ſurdeſolida verbi
              <lb/>
            gratia, quando adduntur 00000. </s>
            <s xml:id="echoid-s13612" xml:space="preserve">fit
              <lb/>
            multiplicatio per 100000. </s>
            <s xml:id="echoid-s13613" xml:space="preserve">id eſt, per
              <lb/>
              <note position="right" xlink:label="note-316-05" xlink:href="note-316-05a" xml:space="preserve">
                <lb/>
              1. # 10. # 100. # 1000. # 10000. # 100000.
                <lb/>
              </note>
            ſurdeſolidum radicis 10. </s>
            <s xml:id="echoid-s13614" xml:space="preserve">Et propor-
              <lb/>
            tio 100000. </s>
            <s xml:id="echoid-s13615" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13616" xml:space="preserve">eſt quintuplicata proportionis 10. </s>
            <s xml:id="echoid-s13617" xml:space="preserve">ad 1. </s>
            <s xml:id="echoid-s13618" xml:space="preserve">vt in appoſito exem-
              <lb/>
            plo apparet, &</s>
            <s xml:id="echoid-s13619" xml:space="preserve">c.</s>
            <s xml:id="echoid-s13620" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div822" type="section" level="1" n="280">
          <head xml:id="echoid-head306" xml:space="preserve">PROBL. 16. PROPOS. 21.</head>
          <p>
            <s xml:id="echoid-s13621" xml:space="preserve">RADICEM cuiuſque generis ex data minutia extrahere.</s>
            <s xml:id="echoid-s13622" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13623" xml:space="preserve">
              <emph style="sc">In</emph>
            minutiis extrahenda eſt radix eiuſdem appellationis cum radice, quæ
              <lb/>
              <note position="left" xlink:label="note-316-06" xlink:href="note-316-06a" xml:space="preserve">Extractio ra-
                <lb/>
              dicum ex mi-
                <lb/>
              nutiis.</note>
            quæritur, tum ex numeratore, tum ex denominatore. </s>
            <s xml:id="echoid-s13624" xml:space="preserve">Ita enim fiet fractio, quæ
              <lb/>
            radix eſt propoſitæ minutiæ. </s>
            <s xml:id="echoid-s13625" xml:space="preserve">Vtradix quadrata minutiæ {4/9}. </s>
            <s xml:id="echoid-s13626" xml:space="preserve">eſt {2/3}. </s>
            <s xml:id="echoid-s13627" xml:space="preserve">Et radix cubica
              <lb/>
            minutiæ {8/27}. </s>
            <s xml:id="echoid-s13628" xml:space="preserve">eſt ſimiliter {2/3
              <unsure/>
            }. </s>
            <s xml:id="echoid-s13629" xml:space="preserve">Et radix Zenſizenſica minutiæ {30/81}. </s>
            <s xml:id="echoid-s13630" xml:space="preserve">eſt quo que {@/@}. </s>
            <s xml:id="echoid-s13631" xml:space="preserve">Et
              <lb/>
            radix ſurdeſolida minutiæ {32/243}. </s>
            <s xml:id="echoid-s13632" xml:space="preserve">eſt pari ratione {@/@}. </s>
            <s xml:id="echoid-s13633" xml:space="preserve">& </s>
            <s xml:id="echoid-s13634" xml:space="preserve">ſic de aliis.</s>
            <s xml:id="echoid-s13635" xml:space="preserve"/>
          </p>
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