Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]

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THEOR. ARITH.
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            <div xml:id="echoid-div234" type="math:theorem" level="3" n="122">
              <p>
                <s xml:id="echoid-s1092" xml:space="preserve">
                  <pb o="89" rhead="THEOR. ARITH." n="95" file="0095" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0095"/>
                autem corporis
                  <var>.M.</var>
                miſti
                  <reg norm="vt" type="punctuation">.vt</reg>
                .216. argentei verò
                  <var>.A.</var>
                vt .156. detrahatur nunc pon-
                  <lb/>
                dus
                  <var>.A.</var>
                ex pondere
                  <var>.V</var>
                . </s>
                <s xml:id="echoid-s1093" xml:space="preserve">Reliquum erit .78. quod vocetur prima differentia ſeruan-
                  <lb/>
                da, dematur etiam pondus
                  <var>.M.</var>
                ex pondere
                  <var>.V.</var>
                reliquum erit .18. pro ſecunda diffe-
                  <lb/>
                rentia, etiam ſeruanda, multiplicetur poſteà pondus
                  <var>.A.</var>
                per ſecundam differen-
                  <lb/>
                tiam, productum verò diuidatur per primam differentiam. </s>
                <s xml:id="echoid-s1094" xml:space="preserve">Vnde in præſenti exem
                  <lb/>
                plo proueniet nobis .36. quiquidem prouentus erit quantitas argenti ipſius corpo-
                  <lb/>
                ris miſti
                  <var>.M.</var>
                quo etiam detracto ex pondere totali ipſius
                  <var>.M.</var>
                reliquum erit quanti-
                  <lb/>
                tas auri eius corporis, hoc eſt .180.</s>
              </p>
              <p>
                <s xml:id="echoid-s1095" xml:space="preserve">In cuius operationis ſpeculatione, aliquid natura ſua prius cognitum præcedere
                  <lb/>
                oportet hoc eſt, quod omnia corpora omogenea eandem proportionem obtinent
                  <lb/>
                inter quantitates, quam inter pondera. </s>
                <s xml:id="echoid-s1096" xml:space="preserve">Quo ſuppoſito denotetur corpus
                  <var>.A.</var>
                li-
                  <lb/>
                nea
                  <var>.o.a.</var>
                corpus autem
                  <var>.V.</var>
                linea
                  <var>.o.c.</var>
                & corpus
                  <var>.M.</var>
                linea
                  <var>.e.u</var>
                : ſed
                  <var>.e.o.</var>
                ſignificet par-
                  <lb/>
                tem argenti, et
                  <var>.o.u.</var>
                partem auri in corpore miſto
                  <var>.M.</var>
                vnde ex communi conceptu
                  <lb/>
                habebimus
                  <var>.o.e.</var>
                æqualem
                  <var>.u.c.</var>
                cum ex hypotheſi
                  <var>.e.u.</var>
                æqualis ſit
                  <var>.o.c.</var>
                et
                  <var>.a.o.</var>
                ſimiliter.
                  <lb/>
                </s>
                <s xml:id="echoid-s1097" xml:space="preserve">Significetur poſteà pondus
                  <var>.a.o.</var>
                ab
                  <var>.f.</var>
                & pondus
                  <var>.e.u.</var>
                ab
                  <var>.b.x.</var>
                & pondus
                  <var>.o.c.</var>
                ab
                  <var>.f.g.</var>
                pon
                  <lb/>
                dus verò
                  <var>.o.e.</var>
                ab
                  <var>.b.</var>
                pondus autem
                  <var>.o.u.</var>
                ab
                  <var>.x.</var>
                pondus enim
                  <var>.u.c.</var>
                ab
                  <var>.b.d.</var>
                et
                  <var>.g.</var>
                ſit diffe-
                  <lb/>
                rentia, qua
                  <var>.f.g.</var>
                maior eſt .f: et
                  <var>.d.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/>
                differentia qua
                  <var>.b.d.</var>
                maior eſt
                  <var>.b</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s1098" xml:space="preserve">Vnde ex ratione omogeneitatis ea
                  <lb/>
                dem proportio erit
                  <var>.a.o.</var>
                ad
                  <var>.e.o.</var>
                vt
                  <var>.
                    <lb/>
                  f.</var>
                ad
                  <var>.b.</var>
                et
                  <var>.o.c.</var>
                ad
                  <var>.u.c.</var>
                quæ
                  <var>.x.b.d.</var>
                ſeu
                  <lb/>
                  <var>f.g.</var>
                (quodidem eſt) ad
                  <var>.b.d</var>
                . </s>
                <s xml:id="echoid-s1099" xml:space="preserve">Quare
                  <lb/>
                ex
                  <ref id="ref-0013">.11. quinti
                    <reg norm="eadem" type="context">eadẽ</reg>
                  </ref>
                erit proportio
                  <var>.
                    <lb/>
                  f.</var>
                ad
                  <var>.b.</var>
                vt
                  <var>.f.g.</var>
                ad
                  <var>.b.d.</var>
                & permutan-
                  <lb/>
                do ita erit
                  <var>.f.</var>
                ad
                  <var>.f.g.</var>
                vt
                  <var>.b.</var>
                ad
                  <var>.b.d.</var>
                &
                  <lb/>
                ſeparando ita
                  <var>.f.</var>
                ad
                  <var>.g.</var>
                vt
                  <var>.b.</var>
                ad
                  <var>.d</var>
                . </s>
                <s xml:id="echoid-s1100" xml:space="preserve">Sed
                  <var>.g.</var>
                cognita nobis eſt, vt differentia in
                  <lb/>
                ter
                  <var>.f.</var>
                g, et
                  <var>.f</var>
                : cognita nobis eſt etiam
                  <var>.f</var>
                : cognoſcimus itidem
                  <var>.d.</var>
                vt differentiam inter
                  <var>.
                    <lb/>
                  x.b.d.</var>
                et
                  <var>.b.x.</var>
                quapropter cognoſcemus
                  <var>.b.</var>
                ex .20. ſeptimi Eucli. & ſic
                  <var>.x.</var>
                reſiduum.
                  <lb/>
                </s>
                <s xml:id="echoid-s1101" xml:space="preserve">ex
                  <var>.b.x</var>
                .</s>
              </p>
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                <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a">
                  <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0095-01"/>
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              </div>
            </div>
            <div xml:id="echoid-div236" type="math:theorem" level="3" n="123">
              <head xml:id="echoid-head141" xml:space="preserve">THEOREMA
                <num value="123">CXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1102" xml:space="preserve">NVNC ex methodo præcedentis propoſiti deuenire poſſumus in cognitio-
                  <lb/>
                nem veræ quantitatis auri, & argenti confuſi in corona Hieronis conſtituen-
                  <lb/>
                do primum duo corpora ſimplicia æqualia inter ſe, & coronæ hoc modo videlicet,
                  <lb/>
                immergendo coronam, ſeu corpus miſtum in aliquod vas aqua plenum, & diligen-
                  <lb/>
                ter colligere aquam, quæ ex eo effundetur, poſteà verò oportet, aliud vas inuenire
                  <lb/>
                præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-
                  <lb/>
                ſteà tantum argenti, quantum ſieri poteſt, vnde vnumquodque horum duorum cor
                  <lb/>
                porum ſimplicium æquale erit mixto, ſeu coronæ, & ſic quod dictum eſt in præce-
                  <lb/>
                cedenti theoremate exequemur.</s>
              </p>
            </div>
            <div xml:id="echoid-div237" type="math:theorem" level="3" n="124">
              <head xml:id="echoid-head142" xml:space="preserve">THEOREMA
                <num value="124">CXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1103" xml:space="preserve">SED vt breuiori methodo idem præſtemus, quod in antecedenti propoſito di-
                  <lb/>
                ctum eſt, quædam theoremata præmittenda ſunt, videlicet quòd quotíeſcunque
                  <lb/>
                fuerint tria corpora, quorum duo inuicem æqualia ſint in quantitate, ſed diuerſa- </s>
              </p>
            </div>
          </div>
        </div>
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