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="id003256">
                <pb pagenum="188" xlink:href="015/01/207.jpg"/>
              traho 2, reliquum remanet 4 tertius numerus. </s>
              <s id="id003257">Item uolo quar­
                <lb/>
              tum, duplico 4 fit 8, detraho 3 remanet 5 quartus numerus: item
                <lb/>
              uolo minorem 3 & 2, duplico 2 fit 4, detraho 3 remanet 1, ſi autem
                <lb/>
              uellem minorem uno, non poſſet, quia eſſet nihil, ſed creſcendo
                <lb/>
              poteſt extendi in infinitum, ita capio 2, & <02> 10, duplico <02> 10, fit <02>
                <lb/>
              40, detraho 2, remanet <02> 40 m: 2, & ita ſi uolo quartum numerum,
                <lb/>
              duplico <02> 40 m: 2 fit <02> 160 m: 4, detrahe <02> 10 ex <02> 160 m: 4, re­
                <lb/>
              manet <02> 90 m:4, & ita 2 <02> 10 <02> 40 m: 2, & <02> 90 m: 4, ſunt in con­
                <lb/>
              tinua proportione arithmetica, & ita poteſt extendi in infini­
                <lb/>
              tum. </s>
              <s id="id003258">Sed ſi uellem unum, aut duos, aut tres terminos, uel quouis
                <lb/>
              medio 5 arithmeticæ, diuido differentiam per 1 p:numero termi­
                <lb/>
              norum, & partes addo minori numero. </s>
              <s id="id003259">Exemplum, uolo tres nu­
                <lb/>
              meros medios inter 2 & 7 in continua proportione arithmeti­
                <lb/>
              ca, detraho 2 à 7 remanet 5, diuido 5 per 1 p: quam 3, id eſt per 4,
                <lb/>
              exit 1 1/4, adde ergo 1 1/4 ad 2 fit 3 1/4 primus terminus, cui adde iterum
                <lb/>
              1 1/4 fit 4 1/2 ſecundus terminus, cui adde iterum 1 1/4 fit 5 3/4 tertius
                <lb/>
              numerus: fient ergo quinque termini, hoc modo in continua pro­
                <lb/>
              portione arithmetica 23 1/4 4 1/2 5 3/4 & 7. Rurſus uolo totidem, uolo
                <lb/>
              inter 2 & <02> 32, detraho 2 ex <02> 32 remanet <02> 32 m: 2, diuido per 4,
                <lb/>
              qui eſt 1 p: numero terminorum, exit <02> 2 m: 1/2, addo ergo <02> 2 m:
                <lb/>
              1/2 ad 2 fit 1 1/2, p: <02> 2 primus terminus, cui iterum addo <02> 2 m: 1/2 fit
                <lb/>
              <02> 8 p:1, ſecundus terminus, cui etiam addo <02> 2 m: 1/2 fit <02> 18 m:
                <lb/>
              1/2, & ita habes tres terminos medios in continua proportione
                <lb/>
              arithmetica inter 2 & <02> 32, & ita ſi uelles quatuor terminos, diui­
                <lb/>
              deres differentiam per 5, & ſi uelles quinque, diuideres per ſex. </s>
              <s id="id003260">&
                <lb/>
              ita de alijs quibuſcunque.</s>
            </p>
            <p type="margin">
              <s id="id003261">
                <margin.target id="marg595"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              _{m}.</s>
            </p>
            <p type="margin">
              <s id="id003262">
                <margin.target id="marg596"/>
              D
                <emph type="italics"/>
              iff,
                <emph.end type="italics"/>
              20.</s>
            </p>
            <p type="main">
              <s id="id003263">Pro Geometrica proponantur, gratia exempli, 2 & 4, ſi uelim in
                <lb/>
              continua proportione tertium, duco 4 in ſemet fit 16, diuido per 2
                <lb/>
              exit 8. & ſi uelles quartum duc 8 in ſe fit 64, diuide per 4 exit 16
                <lb/>
              quartus terminus, & ita in infinitum, & ſi uelles minorem 2, duc 2
                <lb/>
              in ſe fit 4, diuide 4 per 4 exit 1 tertius terminus, & ita ſi uelles mino­
                <lb/>
              rem. </s>
              <s id="id003264">duc 1 in ſe fit 1, diuide per 2 exit 1/2 quartus terminus, & ita ha­
                <lb/>
              bes quoſuis terminos, & eſt ſimilis arithmeticæ hæc operatio, ſed
                <lb/>
              in arithmetica duplicamus unum terminum, & detrahimus alium:
                <lb/>
              in geometrica multiplicamus unum terminum ad productum, &
                <lb/>
              diuidimus per alium. </s>
              <s id="id003265">Et ſi uelim terminum in continua proportio­
                <lb/>
              ne 2 & <02> 10, duco eodem modo <02> 10 in ſe fit 10, diuido per 2 fit 5
                <lb/>
              tertius terminus, & uelim quartum, duco 5 in ſe fit 25, diuido per <02>
                <lb/>
              10 exit <02> 62 1/2 quartus terminus.</s>
            </p>
            <p type="main">
              <s id="id003266">Et ſi uelles plures terminos medios in proportione geometrica, de
                <lb/>
              ducito maius extremum in ſe
                <expan abbr="ſecundũ">ſecundum</expan>
                <expan abbr="denominationẽ">denominationem</expan>
                <expan abbr="inferiorẽ">inferiorem</expan>
              , id </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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