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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            <p type="main">
              <s id="id002694">
                <pb pagenum="151" xlink:href="015/01/170.jpg"/>
              cit 121, quibus diuiſis per 6 ſupereſt 1. Quod etiam ſic demonſtratur
                <lb/>
              de 5, & compoſitis à 5, nam diuiſo 5 in 3 & 2, quadratum eius
                <expan abbr="cõpo­nitur">compo­
                  <lb/>
                nitur</expan>
              ex duplo 3 in 2, in quo nihil ſupereſt, ſi diuidatur per 6, & ex
                <lb/>
              quadrato 3, quòd eſt 9, in quo ſupereſt 3, & ex quadrato 2 quod eſt </s>
            </p>
            <p type="main">
              <s id="id002695">
                <arrow.to.target n="marg520"/>
                <lb/>
              4, ſed iunctis 4 & 3, & abiecto 6 ſupereſt 1, ergo 5 in 5
                <expan abbr="ductũ">ductum</expan>
              , & diui
                <lb/>
              ſo producto relinquitur 1. Et ſimiliter capio 17, et
                <expan abbr="componit̃">componitur</expan>
              ex 12 &
                <lb/>
              5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil ſu­
                <lb/>
              pereſt, & duplo 5 in 12, in quo
                <expan abbr="etiã">etiam</expan>
              nihil ſupereſt, ſi diuidatur per 6:
                <lb/>
              & ex quadrato 5, in quo ſupereſt 1, ergo in nullo numero
                <expan abbr="cõpoſito">compoſito</expan>
                <lb/>
              ex 5 & 6, uel compoſitis ex 6, poterit produci numerus, qui diuiſus
                <lb/>
              per 6 relinquat 5, igitur neque talis numerus potérit
                <expan abbr="cõponi">componi</expan>
              ex duo­
                <lb/>
              bus quadratis, in quib. </s>
              <s id="id002696">ſuperſit 5 & 1, quia nullus eſt, in quo ſuper­
                <lb/>
              ſit 5 facta diuiſione per 6. Ex quo colligitur una regula: quod ſi quis
                <lb/>
              dicat multiplicaui 27 in ſe, et diuiſi per 13, uellem ſcire quid ſupereſt,
                <lb/>
              dico quod ſine multiplicatione et diuiſione poteris hoc ſcire ex de­
                <lb/>
              monſtratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in ſe
                <lb/>
              fit 1: dices ergo, quod ſupererit 1, & ita ſi ducerem 28 in ſe, & diuide­
                <lb/>
              rem per 11, dico quod ſupererit 3, nam diuiſo 28 per 11, relinquitur
                <lb/>
              6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum eſt, & tantum
                <lb/>
                <expan abbr="relinquit̃">relinquitur</expan>
              ducto 28 in ſe & fit 784, & diuiſo per 11. Reuertendo ergo
                <lb/>
              ad propoſitum, pater quod ex duobus tantum numeris imparibus
                <lb/>
              quadratis poteſt conflari ille numerus,
                <expan abbr="quorũ">quorum</expan>
              radices diuiſæ per 6
                <lb/>
              relinquunt 3. Sed de paribus uel ſupereſt 2 uel 4 uel nihil, ſed
                <expan abbr="q̃dra­tum">quadra­
                  <lb/>
                tum</expan>
              2 eſt 4, &
                <expan abbr="q̃dratum">quadratum</expan>
              4 diuiſum per 6 etiam relinquit 4, ergo neque
                <lb/>
              ex duobus numeris, in quibus ſuperſint 2, neque in quibus ſuperſint
                <lb/>
              4, neque in quibus ſuperſint in uno 2, in altero 4
                <expan abbr="poterũt">poterunt</expan>
              quadrata, in
                <lb/>
              quibus ſemper ſupererit 4, & iuncta faciunt 8, in quod ̊ſupereſt 2,
                <expan abbr="cõ">con</expan>
              fla­
                <lb/>
              re
                <expan abbr="numerũ">numerum</expan>
                <expan abbr="dictũ">dictum</expan>
              ſeu
                <expan abbr="quæſitũ">quæſitum</expan>
              , qui poſsit diuidi per 6: neque ex
                <expan abbr="q̃d">quad</expan>
              . </s>
              <s id="id002697">
                <expan abbr="duo­rũ">duo­
                  <lb/>
                rum</expan>
                <expan abbr="numẽrorũ">numerorum</expan>
              , in
                <expan abbr="quorũ">quorum</expan>
              altero nihil ſuperſit in reliquo ſuperſit 2 uel
                <lb/>
              4, quia in aggregato
                <expan abbr="q̃dratorũ">quadratorum</expan>
              ſemper ſupererit 4. Ergo relinqui­
                <lb/>
              tur quod ille numerus componetur ex duobus quadratis, uel impa
                <lb/>
              ribus, quorum latera diuiſa per 6 relinquunt 3, uel ex duobus pari­
                <lb/>
              bus, quorum latera diuiſa per 6 nihil relinquant. </s>
              <s id="id002698">Oportet igitur
                <lb/>
              inuenire duos tales numeros quadratos numerorum imparium, in
                <lb/>
              quibus ſuperſit 3, ſi diuidantur per 6, aut parium in quibus nihil ſu­
                <lb/>
              perſit, quorum aggregato diuiſo per 6 prodeat numerus
                <expan abbr="q̃dratus'">quadratus'</expan>
              .</s>
            </p>
            <p type="margin">
              <s id="id002699">
                <margin.target id="marg520"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              4.
                <emph type="italics"/>
              ſecun
                <lb/>
              di
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s id="id002700">His uiſis dico, quod conſtat radices talium numerorum opor­
                <lb/>
              tere eſſe in imparibus per additionem 6 incipiendo à 3, ut ſint
                <lb/>
              3. 9. 15. 21. 27. 33. 39. 45. 51. & ſic deinceps: in paribus au­
                <lb/>
              tem per additionem eiuſdem 6 incipiendo à 6, uelut 6. 12.
                <lb/>
              18. 24. 30. 36. 42. 48. 54. 60. Dico ergo quod diui­
                <lb/>
              ſo numero illo compoſito per 6 in imparibus exibit numerus, </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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