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>
            <pb pagenum="147" xlink:href="015/01/166.jpg"/>
            <p type="head">
              <s id="id002599">SCHOLIVM</s>
            </p>
            <p type="main">
              <s id="id002600">Ex hoc pater qualiter ex uera demonſtratione ſenſu oſtenſa per­
                <lb/>
              uenimus ad quotquot imaginando, inde intellectu abiectis condi­
                <lb/>
              tionibus non neceſſarijs facimus infinitum & uniuerſale. </s>
              <s id="id002601">Demum
                <lb/>
              ſine artis ſpecialis auxilio oſtendimus theorema uniuerſale (quod
                <lb/>
              etiam poterat oſtendi Geometricè, ſed longè pulchrius eſt, ac ſubli­
                <lb/>
              mius per
                <foreign lang="grc">περιλαμπουσιν</foreign>
              , qua hoc ipſo infinita alia docemus generaliter
                <lb/>
              per ſimplicem
                <expan abbr="comprehẽſionem">comprehenſionem</expan>
              oſtendere) ſcilicet quod à quouis
                <lb/>
              puncto peripherię circuli, cuius ſemidiameter eſt media proportio­
                <lb/>
              ne inter totam extenſam à centro uſque exterius, & partem quæ' eſt à
                <lb/>
              centro ad punctum deſcriptum ſub proportione continua
                <expan abbr="datarũ">datarum</expan>
                <lb/>
              linearum lineæ ductæ ex eo ad punctum exterius, & punctum de­
                <lb/>
              ſcriptum ſunt in proportione datarum linearum.</s>
            </p>
            <p type="main">
              <s id="id002602">Propoſitio centeſima quinquageſima quinta.</s>
            </p>
            <p type="main">
              <s id="id002603">
                <expan abbr="Quadratorũ">Quadratorum</expan>
                <expan abbr="numerorũ">numerorum</expan>
              proportionem &
                <expan abbr="inuentionẽ">inuentionem</expan>
                <expan abbr="cõſiderare">conſiderare</expan>
              .</s>
            </p>
            <figure id="id.015.01.166.1.jpg" xlink:href="015/01/166/1.jpg" number="172"/>
            <p type="main">
              <s id="id002604">Primùm oportet ſcire eſſe tres naturales
                <lb/>
              numerorum ſeries, primam Euclidis iuxta </s>
            </p>
            <p type="main">
              <s id="id002605">
                <arrow.to.target n="marg514"/>
                <lb/>
              quamuis
                <expan abbr="proportionẽ">proportionem</expan>
              , in qua unum & ter­
                <lb/>
              tius & quintus, & ita uno ſemper intermiſ­
                <lb/>
              ſo ſunt quadrati. </s>
              <s id="id002606">Primus quo que. </s>
              <s id="id002607">1. unum &
                <lb/>
              quartus & ſeptimus & ita duobus intermiſsis ſunt cubi. </s>
              <s id="id002608">In ſecun­
                <lb/>
              do ordine eſt naturalis ſeries numerorum, ex qua colligitur alia, &
                <lb/>
              ex illa bini quilibet ſe ſequentes conſtituunt numerum
                <expan abbr="quadratũ">quadratum</expan>
              .
                <lb/>
              </s>
              <s id="id002609">In tertia numeri impares, qui ſemper collati efficiunt quadratum.</s>
            </p>
            <p type="margin">
              <s id="id002610">
                <margin.target id="marg514"/>
              E
                <emph type="italics"/>
                <expan abbr="xemplũ">xemplum</expan>
                <emph.end type="italics"/>
              1.</s>
            </p>
            <figure id="id.015.01.166.2.jpg" xlink:href="015/01/166/2.jpg" number="173"/>
            <p type="main">
              <s id="id002611">Sit ergo propoſitus numerus cui uelim
                <lb/>
              addere quadratum numerum, ut fiat qua­
                <lb/>
                <arrow.to.target n="marg515"/>
                <lb/>
              dratus totus, accipe numerum quadratum
                <lb/>
              minorem illo quem uis, & detrahe à propo
                <lb/>
              ſito numero ſeu quadrato ſeu non reſidu­
                <lb/>
                <arrow.to.target n="marg516"/>
                <lb/>
              um, diuide per duplum <02> quadrati quod
                <lb/>
              detraxiſti, q̊d exit duc in ſe fiet quadratus numerus, idem que additus
                <lb/>
              numero propoſito, faciet quadratum. </s>
              <s id="id002612">Velut capio 16 qui eſt qua­
                <lb/>
              dratus, aufero 9 quadratum
                <expan abbr="minorẽ">minorem</expan>
              relinquitur 7, diuido per 6 du­
                <lb/>
              plum <02> 9, exit 1 1/6 quadratum eius eſt 1 13/36 qui additus ad 16 facit 17 13/36
                <lb/>
                <expan abbr="quadratũ">quadratum</expan>
              cuius <02> eſt 4 1/6.</s>
            </p>
            <p type="margin">
              <s id="id002613">
                <margin.target id="marg515"/>
              E
                <emph type="italics"/>
                <expan abbr="xemplũ">xemplum</expan>
                <emph.end type="italics"/>
              2.</s>
            </p>
            <p type="margin">
              <s id="id002614">
                <margin.target id="marg516"/>
              E
                <emph type="italics"/>
                <expan abbr="xemplũ">xemplum</expan>
                <emph.end type="italics"/>
              3.</s>
            </p>
            <p type="main">
              <s id="id002615">Ex hoc patet propoſito quouis numero
                <expan abbr="q̃drato">quadrato</expan>
              modus inuenien­
                <lb/>
                <arrow.to.target n="marg517"/>
                <lb/>
              di infinitos numeros quadratos qui
                <expan abbr="cũ">cum</expan>
              illo iuncti facient
                <expan abbr="quadratũ">quadratum</expan>
              .</s>
            </p>
            <p type="margin">
              <s id="id002616">
                <margin.target id="marg517"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 1.</s>
            </p>
            <p type="head">
              <s id="id002617">SCHOLIVM.</s>
            </p>
            <p type="main">
              <s id="id002618">Poſſem adducere demonſtrationes omnium
                <expan abbr="horũ">horum</expan>
              , ſed reddere­
                <lb/>
              tur res longa
                <expan abbr="cũ">cum</expan>
              ſint manifeſtę ex ſeptimo octauo & nono Euclidis.
                <lb/>
              </s>
              <s id="id002619">Exemplum ſecundum capio modò 14 qui non eſt quadratus, aufe­
                <lb/>
              ro 9, remanet 5, diuido per 6 duplum <02> 9 exit 5/6
                <expan abbr="quadratũ">quadratum</expan>
              eius eſt 25/36 </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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