Cardano, Geronimo, Opvs novvm de proportionibvs nvmerorvm, motvvm, pondervm, sonorvm, aliarvmqv'e rervm mensurandarum, non solùm geometrico more stabilitum, sed etiam uarijs experimentis & observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in V libros digestum. Praeterea Artis Magnae, sive de regvlis algebraicis, liber vnvs abstrvsissimvs & inexhaustus planetotius Ariothmeticae thesaurus ... Item De Aliza Regvla Liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum Geometricas quantitates inquirentis ...

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="id001067">
                <pb pagenum="57" xlink:href="015/01/076.jpg"/>
              ab eadem analogæ, erit proportio tertiæ unius ordinis ad tertiam
                <lb/>
              alterius, ut ſecundæ ad ſecundam duplicata, & quartæ ad quartam
                <lb/>
              triplicata, quintæ ad quintam quadruplicata, at que ſic de alijs.
                <lb/>
                <arrow.to.target n="marg195"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001068">
                <margin.target id="marg195"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              _{m}.</s>
            </p>
            <p type="main">
              <s id="id001069">Sint quantitates b c d e f, ab a in continua proportio­
                <lb/>
                <figure id="id.015.01.076.1.jpg" xlink:href="015/01/076/1.jpg" number="73"/>
                <arrow.to.target n="table14"/>
                <lb/>
              ne, & aliæ totidem g h k l m, dico quod proportio h c eſt
                <lb/>
              duplicata ei, quæ eſt g ad b, & k ad d triplicata, & l ad e
                <lb/>
              quadruplicata, & ſic deinceps, ſumatur enim unum, & ab </s>
            </p>
            <table>
              <table.target id="table14"/>
              <row>
                <cell/>
                <cell>a</cell>
                <cell/>
              </row>
              <row>
                <cell>b</cell>
                <cell/>
                <cell>g</cell>
              </row>
              <row>
                <cell>c</cell>
                <cell/>
                <cell>h</cell>
              </row>
              <row>
                <cell>d</cell>
                <cell/>
                <cell>k</cell>
              </row>
              <row>
                <cell>e</cell>
                <cell/>
                <cell>l</cell>
              </row>
              <row>
                <cell>f</cell>
                <cell/>
                <cell>m</cell>
              </row>
              <row>
                <cell/>
                <cell>n</cell>
                <cell/>
              </row>
              <row>
                <cell>o</cell>
                <cell/>
                <cell>t</cell>
              </row>
              <row>
                <cell>p</cell>
                <cell>
                  <foreign lang="grc">α</foreign>
                </cell>
                <cell>u</cell>
              </row>
              <row>
                <cell>q</cell>
                <cell>
                  <foreign lang="grc">β γ</foreign>
                </cell>
                <cell>x</cell>
              </row>
              <row>
                <cell>z</cell>
                <cell/>
                <cell>y</cell>
              </row>
              <row>
                <cell>s</cell>
                <cell/>
                <cell>z</cell>
              </row>
            </table>
            <p type="main">
              <s id="id001070">
                <arrow.to.target n="marg196"/>
                <lb/>
              eo o p q r s in proportione b ad a, & t u x y z in propor­
                <lb/>
              tione g ad a, erit igitur p quadratum o, & u quadratum t,
                <lb/>
              & q cubus o, & x cubus t, & ita de alijs: ergo proportio
                <lb/>
                <arrow.to.target n="marg197"/>
                <lb/>
              n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæ t
                <lb/>
              ad o, & poteſt etiam demonſtrari generaliter ultra qua­
                <lb/>
                <arrow.to.target n="marg198"/>
                <lb/>
              dratum, & cubum: nam ſi ducatur t in o, fiat que
                <foreign lang="grc">α</foreign>
              erit, pro­
                <lb/>
              portio enim ad
                <foreign lang="grc">α</foreign>
              eadem quæ t ad o, & proportio a ad p,
                <lb/>
              ut t ad o, igitur per diffinitionem proportionis duplicatæ
                <lb/>
                <arrow.to.target n="marg199"/>
                <lb/>
              poſitam in quinto libro ab Euclide u ad p duplicata ei,
                <lb/>
              quæ t ad o, & ſimiliter ex t in p fit
                <foreign lang="grc">β</foreign>
              ex o in u,
                <foreign lang="grc">γ</foreign>
              eruntque
                <lb/>
                <arrow.to.target n="marg200"/>
                <lb/>
              q
                <foreign lang="grc">β γ</foreign>
              x in continua proportione per eandem. </s>
              <s id="id001071">Quia ergo propor­
                <lb/>
              tio q ad
                <foreign lang="grc">β</foreign>
              eſt ut o ad t, patet, quod x ad q eſt triplicata ei, quæ eſt t ad
                <lb/>
              o, & ita de reliquis, cum ergo proportio p ad o ſit, ut e ad b, & o ad
                <lb/>
                <arrow.to.target n="marg201"/>
                <lb/>
              n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, ſequitur ut ſit t ad a,
                <lb/>
              ut g ad b, & u ad p, ut h ad c, igitur cum ſit ut u ad p duplicata ei, quę
                <lb/>
              eſt t ad o erit h ad e, duplicata ei quæ eſt g ad b, & ita de reliquis, &
                <lb/>
              noǹ refert, ſeu dicas u ad p duplicatam ei, quæ eſt t ad o, ſeu dicas p
                <lb/>
                <arrow.to.target n="marg202"/>
                <lb/>
              ad u duplicatam ei, quæ eſt o ad t. </s>
              <s id="id001072">Aliter & euidentius in duabus
                <lb/>
              ſoleo demonſtrare: cum enim ſit e & h duplicata ei quæ eſt b & g
                <lb/>
              ad a, ut ſupra, & quadrati b ad quadratum a, & quadrati g ad qua­
                <lb/>
                <arrow.to.target n="marg203"/>
                <lb/>
              dratum a duplicata his quæ b & g ad a erunt b & g quadratorum
                <lb/>
              ad quadratum a, uelut c & h ad a. </s>
              <s id="id001073">Et conuertendo qua­
                <lb/>
                <arrow.to.target n="table15"/>
                <lb/>
              drati a ad quadratum g, ut a ad h, conſtituantur ergo
                <lb/>
                <figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg" number="74"/>
              hic & erit quadrati b ad
                <expan abbr="quadratũ">quadratum</expan>
              g, ita c ad h: ſed qua­
                <lb/>
              drati b ad quadratum g, ut b ad g proportio duplicata
                <lb/>
              igitur e ad h, ut b ad g duplicata.</s>
            </p>
            <p type="margin">
              <s id="id001074">
                <margin.target id="marg196"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              8.
                <emph type="italics"/>
              noni
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              le.
                <emph.end type="italics"/>
              & 22. & 23.
                <emph type="italics"/>
              octaui.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001075">
                <margin.target id="marg197"/>
              V
                <emph type="italics"/>
              ide per
                <emph.end type="italics"/>
              23. P
                <emph type="italics"/>
              etit.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001076">
                <margin.target id="marg198"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              23.
                <emph type="italics"/>
              ſex ti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              & 33.
                <emph type="italics"/>
              undeci­mi.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001077">
                <margin.target id="marg199"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              17.
                <emph type="italics"/>
              ſe­ptimi
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001078">
                <margin.target id="marg200"/>
              D
                <emph type="italics"/>
              iff.
                <emph.end type="italics"/>
              10.</s>
            </p>
            <p type="margin">
              <s id="id001079">
                <margin.target id="marg201"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              24.
                <emph type="italics"/>
              quinti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001080">
                <margin.target id="marg202"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              10
                <emph type="italics"/>
              diff. </s>
              <s id="id001081">quinti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id001082">
                <margin.target id="marg203"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              20.
                <emph type="italics"/>
              ſex ti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lement.
                <emph.end type="italics"/>
              </s>
            </p>
            <table>
              <table.target id="table15"/>
              <row>
                <cell>
                  <expan abbr="q̃d">quad</expan>
                .</cell>
                <cell>b</cell>
                <cell>e</cell>
              </row>
              <row>
                <cell>
                  <expan abbr="q̃d">quad</expan>
                .</cell>
                <cell>a</cell>
                <cell>a</cell>
              </row>
              <row>
                <cell>
                  <expan abbr="q̃d">quad</expan>
                .</cell>
                <cell>g</cell>
                <cell>h</cell>
              </row>
            </table>
            <p type="main">
              <s id="id001083">Propoſitio ſexageſimaoctaua, collectorum ab Euclide
                <lb/>
              & Archimede.</s>
            </p>
            <p type="main">
              <s id="id001084">Omnis cylindrus cono habenti baſim, & altitudinem eandem
                <lb/>
                <arrow.to.target n="marg204"/>
                <lb/>
              triplus eſt. </s>
              <s id="id001085">Omnis cylindrus ſphæræ habenti eundem magnum
                <lb/>
                <arrow.to.target n="marg205"/>
                <lb/>
              circulum, & altitudinem ſexquialter eſt. </s>
              <s id="id001086">Omnis ſphæra dupla eſt
                <lb/>
                <arrow.to.target n="marg206"/>
                <lb/>
              cono, cuius baſis eſt eius circulus magnus, & altitudo eadem, quæ
                <lb/>
              ſphæræ ipſius. </s>
              <s id="id001087">Omnis ſuperficies ſphæræ quadrupla eſt maiori
                <lb/>
                <arrow.to.target n="marg207"/>
                <lb/>
              ſuo circulo. </s>
              <s id="id001088">Superficies portionis ſphæræ eſt æqualis circulo, cu
                <lb/>
                <arrow.to.target n="marg208"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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