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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            <pb pagenum="185 [=191]" xlink:href="015/01/210.jpg"/>
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              <s id="id003283">Ex hoc patet q̊d in
                <expan abbr="proportionẽ">proportionem</expan>
              Arithmetica & muſica ſemper, ſi
                <lb/>
                <arrow.to.target n="marg597"/>
                <lb/>
              duo termini fuerint numeri, tertius erit numerus, & in Geometrica
                <lb/>
              idem erit, ſi medius & extremus fuerint numeri, erit alter extremus
                <lb/>
              numerus, ſed tamen ſi unus euariet, omnes poterunt eſſe diuerſi.</s>
            </p>
            <p type="margin">
              <s id="id003284">
                <margin.target id="marg597"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003285">Propoſitio centeſima ſeptuageſima ſecunda.</s>
            </p>
            <p type="main">
              <s id="id003286">Proportiones Stiphelij deſcribere.</s>
            </p>
            <p type="main">
              <s id="id003287">Conſiderauit Michael Stiphelius quod ſumpſit à
                <expan abbr="Boẽtio">Boentio</expan>
              , quaſ­
                <lb/>
                <arrow.to.target n="marg598"/>
                <lb/>
              dam inueniri proportiones tribus numeris conſtitutis, quæ in nul­
                <lb/>
              lo trium primorum generum continerentur, ſed quædam tamen
                <lb/>
              geometricis aliæ muſicis aſsimilarentur, prima ergo Geometrica­
                <lb/>
              rum eſt, quoties proportio ſecundæ ad primam fuerit, uelut diffe­
                <lb/>
              rentiæ ſecundæ & primæ ad differentiam ſecundæ & tertiæ. </s>
              <s id="id003288">Velut
                <lb/>
                <arrow.to.target n="marg599"/>
                <lb/>
              capio 2, 4, 5, proportio 4 ad 2 eſt dupla talis eſt 2 differentiæ 4 & 2
                <lb/>
                <arrow.to.target n="marg600"/>
                <lb/>
              ad 1 differentiam 5 & 4, nam in uera proportione Geometrica fit
                <lb/>
              conuerſo modo, quia proportio ſecundæ ad primam eſt, uelut dif­
                <lb/>
              ferentię tertiæ & ſecundæ ad differentiam ſecundæ à prima ut in 4.
                <lb/>
              6. & 9 proportio 6 ad 4 eſt uelut 3 differentiæ 9 ad 6 ad 2 differen­
                <lb/>
              tiam 6 & 4.</s>
            </p>
            <p type="margin">
              <s id="id003289">
                <margin.target id="marg598"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id003290">
                <margin.target id="marg599"/>
              2 1</s>
            </p>
            <p type="margin">
              <s id="id003291">
                <margin.target id="marg600"/>
              2 4 5</s>
            </p>
            <p type="main">
              <s id="id003292">
                <expan abbr="Secũda">Secunda</expan>
              proportio quam ille appellat poſteriorem, eſt in qua pro
                <lb/>
              portio tertij ad ſecundum eſt uelut differentiæ primi & ſecundi ad
                <lb/>
              differentiam ſecundi & tertij: Velut capio 1, 4, 6, proportio 6 ad 4
                <lb/>
                <arrow.to.target n="marg601"/>
                <lb/>
              tertij ſcilicet, & ſecundum eſt uelut 3 differentiæ 4 & 1, ad 2, differen­
                <lb/>
                <arrow.to.target n="marg602"/>
                <lb/>
              tiam 6 & 4, & hæc ſimiliter differt à Geometrica uera in eo quo in
                <lb/>
              Geometrica uera oporteret, ut proportio tertij ad ſecundum eſſet
                <lb/>
              ut differentia tertij & ſecundi ad differentiam ſecundi & primi. </s>
              <s id="id003293">Dif­
                <lb/>
              fert à priore, quoniam in illa differentiæ ſeruant eundem ordinem,
                <lb/>
              quanuis transferantur in hac uerò fit conuerſus modus.</s>
            </p>
            <p type="margin">
              <s id="id003294">
                <margin.target id="marg601"/>
              3 2</s>
            </p>
            <p type="margin">
              <s id="id003295">
                <margin.target id="marg602"/>
              1 4 6</s>
            </p>
            <p type="main">
              <s id="id003296">Tertia eſt ut ſit proportio differentiæ primæ & tertiæ ad diffe­
                <lb/>
              rentiam primæ & ſecundæ, uelut ſecundæ ad primam, in Geometri
                <lb/>
              ca autem eſſet ſicut aggregati ſecundæ & primæ ad ipſam primam,
                <lb/>
              tales ergo quantitates erunt uelut 4, 6, 7, nam proportio 6 ad 4 eſt
                <lb/>
                <arrow.to.target n="marg603"/>
                <lb/>
              uelut 3 differentiæ 4 & 7 ad 2 differentiam 4 & 6.
                <lb/>
                <arrow.to.target n="marg604"/>
                <lb/>
                <arrow.to.target n="marg605"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003297">
                <margin.target id="marg603"/>
              3</s>
            </p>
            <p type="margin">
              <s id="id003298">
                <margin.target id="marg604"/>
              4 6 7</s>
            </p>
            <p type="margin">
              <s id="id003299">
                <margin.target id="marg605"/>
              2</s>
            </p>
            <p type="main">
              <s id="id003300">Quarta proportio ſimilis Geometricæ eſt cum fuerit proportio
                <lb/>
              differentiæ primæ & tertiæ ad differentiam tertiæ & ſecundę, uelut
                <lb/>
              ſecundæ ad primam, uelut in 2, 3, 5 proportio differentiæ 5 & 2 quæ </s>
            </p>
            <p type="main">
              <s id="id003301">
                <arrow.to.target n="marg606"/>
                <lb/>
                <arrow.to.target n="marg607"/>
                <lb/>
              eſt 3 ad differentiam ſecundæ & tertiæ, quæ eſt 2 eſt uelut 3 quantita
                <lb/>
                <arrow.to.target n="marg608"/>
                <lb/>
              tis ſecundæ ad 2 quantitatem primam.</s>
            </p>
            <p type="margin">
              <s id="id003302">
                <margin.target id="marg606"/>
              3</s>
            </p>
            <p type="margin">
              <s id="id003303">
                <margin.target id="marg607"/>
              2 3 5</s>
            </p>
            <p type="margin">
              <s id="id003304">
                <margin.target id="marg608"/>
              2</s>
            </p>
            <p type="main">
              <s id="id003305">Prima
                <expan abbr="autẽ">autem</expan>
                <expan abbr="harmonicarũ">harmonicarum</expan>
              quæ notha eſt nec legitima, hoc modo
                <lb/>
              ſumitur: Vt ſit proportio primæ ad tertiam uelut differentiæ ſecun
                <lb/>
                <arrow.to.target n="marg609"/>
                <lb/>
              dæ & tertiæ ad differentiam ſecundæ & primæ, ueluti capio 6 pri­
                <lb/>
                <arrow.to.target n="marg610"/>
                <lb/>
              mam 5 ſecundum 3 tertiam proportio 6 ad 3 eſt dupla ſicut 2 </s>
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          </chap>
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