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="id002674">
                <pb pagenum="150" xlink:href="015/01/169.jpg"/>
              integri ex ratione dicta, quia oporteret ut eſſent ambo impares aut
                <lb/>
              pares, & ſic
                <expan abbr="differrẽt">differrent</expan>
              numero pari, ergo oporteret ut eſſet unus me­
                <lb/>
              dius numerus
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002675">ſunt & alię rationes, ſed neque unus poſſet eſſe inte
                <lb/>
              ger, & alius fractus,
                <expan abbr="nõ">non</expan>
              eſſet. </s>
              <s id="id002676">n. </s>
              <s id="id002677">6 numerus integer:
                <expan abbr="relinquit̃">relinquitur</expan>
              ergo ut
                <lb/>
              ſint duo fracti: ſed in numeris fractis
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002678">deductis ad minimas deno
                <lb/>
              minationes
                <expan abbr="operũ">operum</expan>
              , ut tam denominator <08> numerator habeat radi­
                <lb/>
              ces, ergo oportet q̊d hoc ſit in illis, & quia iuncti debent facere inte­
                <lb/>
              gros 6, neceſſe eſt ut denominator ſit unus, &
                <expan abbr="idẽ">idem</expan>
              in utroque, et q̊d nu
                <lb/>
              meratores ſimul iuncti ſint
                <expan abbr="ſexcuplũ">ſexcuplum</expan>
              denominatoris, ſi fracti
                <expan abbr="debẽt">debent</expan>
                <lb/>
              ęquipollere 6, ergo ille denominator
                <expan abbr="cũ">cum</expan>
              ſit
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002679">& numeratores am­
                <lb/>
              bo ſint
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002680">& ſint
                <expan abbr="ſexcuplũ">ſexcuplum</expan>
              denominatoris, oportebit inuenire
                <expan abbr="nu­merũ">nu­
                  <lb/>
                merum</expan>
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002681">qui ductus in 6, faciat
                <expan abbr="numerũ">numerum</expan>
              qui
                <expan abbr="cõponit̃">componitur</expan>
              ex duob. </s>
              <s id="id002682">
                <expan abbr="q̃d">quad</expan>
              .
                <lb/>
              </s>
              <s id="id002683">aut
                <expan abbr="cõponit̃">componitur</expan>
              ęqualiter, ergo proportio medietatis ad
                <expan abbr="medietatẽ">medietatem</expan>
              6, eſt
                <lb/>
              ueluti totius ad 6, ſed totu continet 6 in
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002684">quia ex 6 in
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002685">fit
                <expan abbr="totũ">totum</expan>
              ,
                <lb/>
              ergo ex medietate in
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002686">idem fit medietas, ſed medietas eſt nume­
                <lb/>
              rus
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002687">ergo 3 eſſet numerus
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002688">quod eſt falſum, oportet
                <expan abbr="igit̃">igitur</expan>
              ut nume
                <lb/>
              ri illi ſint inæ quales, & ut 6 diuidatur in duas partes inęquales, hoc
                <lb/>
                <expan abbr="aũt">aut</expan>
              fit diuidendo quemlibet
                <expan abbr="numerũ">numerum</expan>
              parem, qui
                <expan abbr="cõponit̃">componitur</expan>
              ex duob.
                <lb/>
              </s>
              <s id="id002689">numeris
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002690">nam ſi eſſet impar,
                <expan abbr="nõ">non</expan>
              poſſet prodire numerus integer, &
                <lb/>
                <expan abbr="cũ">cum</expan>
              prouenerit numerus
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002691">ille erit
                <expan abbr="quẽ">quem</expan>
              quęrimus,
                <expan abbr="nã">nam</expan>
              diuiſo 6 per to­
                <lb/>
              tum
                <expan abbr="illũ">illum</expan>
              numerum, inde q̊d prouenit multiplicato per numeros
                <expan abbr="q̃d">quad</expan>
              ,
                <lb/>
                <expan abbr="cõponentes">componentes</expan>
              illum
                <expan abbr="numerũ">numerum</expan>
              productum,
                <expan abbr="producunt̃">producuntur</expan>
              partes 6, quæ
                <expan abbr="erũt">erunt</expan>
                <lb/>
              numeri
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002692">quia denominator utriuſque partis ex ſuppoſito eſt nume
                <lb/>
              rus
                <expan abbr="q̃dratus">quadratus</expan>
              , qui multiplicatus eſt per 6, & numeratores ſunt nume
                <lb/>
              ri
                <expan abbr="q̃drati">quadrati</expan>
              , qui
                <expan abbr="cõponebant">componebant</expan>
                <expan abbr="numerũ">numerum</expan>
                <expan abbr="productũ">productum</expan>
              , et tales partes
                <expan abbr="ęquant̃">ęquantur</expan>
                <lb/>
              6, quia numerus productus
                <expan abbr="componit̃">componitur</expan>
              ex numeratoribus, &
                <expan abbr="produ­cit̃">produ­
                  <lb/>
                citur</expan>
              tale
                <expan abbr="cõpoſitum">compoſitum</expan>
              ex 6 in
                <expan abbr="denominatorẽ">denominatorem</expan>
              , & hic eſt diuiſus per deno
                <lb/>
                <expan abbr="minatorẽ">minatorem</expan>
              , ergo prouenit 6, ſi
                <expan abbr="em̃">emm</expan>
              multiplicato 3 in 4 fit 12, diuiſo 12 per
                <lb/>
              4, exit neceſſario idem 3. Pro colligendo ergo numeros omnes, qui
                <lb/>
                <expan abbr="cõponuntur">componuntur</expan>
              ex
                <expan abbr="q̃dratis">quadratis</expan>
              , propones tibi ſeriem
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002693">
                <expan abbr="omniũ">omnium</expan>
              , & inde iun­
                <lb/>
              ges, & diuides per 6, &
                <expan abbr="cũ">cum</expan>
              prodierit
                <expan abbr="q̃dratus">quadratus</expan>
              ,
                <expan abbr="inuenit̃">inuenitur</expan>
              denominator,
                <lb/>
              & numeri
                <expan abbr="cõponentes">componentes</expan>
              ipſum erunt numeratores, et ſuppoſiti deno
                <lb/>
              minatoribus
                <expan abbr="cõſtituent">conſtituent</expan>
              partes. </s>
              <s id="id002694">Vt uerò cognoſcas, ex quibus poſ­
                <lb/>
              ſit componi primum ex imparibus, non oportet aſſumere niſi 135,
                <lb/>
              quia 7 diuiſum per 6 relinquit 1, & 9 diuiſum per 6, relinquit 3, & 35
                <lb/>
              diuiſum per 6 relinquit 5. ergo non poteſt componi numerus im­
                <lb/>
              par, qui diuidatur per 6, ut ſuperſit impar alius quàm 1. 3. 5. ſed 1 & 3
                <lb/>
              & 5, & 5 componunt 4 & 1, & 1 & 3 & 5 componunt 2, ſcilicet abie­
                <lb/>
              cto 6, ergo tales numeri
                <expan abbr="q̃drati">quadrati</expan>
              ſi ſint impares, uel ambo terminan­
                <lb/>
              tur in 3, ut 9 & 81, qui faciunt 90, uel in 1 & 5, ſed nullus numerus
                <lb/>
              quadratus diuiſus per 6 terminatur in 5, quia 1 ductum in ſe produ­
                <lb/>
              cit 1, & 3 pro ducit 3, & 5 pro ducit 1, ut 5 in 5 facit 25, & 11 in 11 </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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