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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THEOREM. ARIT.
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            <div xml:id="echoid-div240" type="math:theorem" level="3" n="126">
              <p>
                <s xml:id="echoid-s1113" xml:space="preserve">
                  <pb o="85" rhead="THEOREM. ARIT." n="97" file="0097" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0097"/>
                pori
                  <var>.a.</var>
                argentei, corpus autem
                  <var>.e.</var>
                  <reg norm="aqueum" type="context">aqueũ</reg>
                æqualis ma-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/>
                gnitudinis corpori
                  <var>.b.</var>
                aurei. </s>
                <s xml:id="echoid-s1114" xml:space="preserve">Tunc dico proportio-
                  <lb/>
                nem ponderis
                  <var>.f.</var>
                ad pondus
                  <var>.e.</var>
                eadem eſſe, quæ pon-
                  <lb/>
                deris
                  <var>.b.</var>
                ad pondus
                  <var>.u.</var>
                vt in præcedenti theoremate
                  <lb/>
                iam dictum eſt, vel quæ ponderis
                  <var>.n.</var>
                ad pondus
                  <var>.a.</var>
                ex
                  <lb/>
                11. quinti Euclidis. </s>
                <s xml:id="echoid-s1115" xml:space="preserve">Proptereà quòe ponderis
                  <var>.
                    <lb/>
                  n.</var>
                ad pondus
                  <var>.a.</var>
                eft vt poderis
                  <var>.b.</var>
                ad pondus
                  <var>.u.</var>
                eo
                  <lb/>
                quòd permutando ponderis
                  <var>.n.</var>
                ad pondus
                  <var>.b.</var>
                eſt vt
                  <lb/>
                ponderis
                  <var>.a.</var>
                ad pondus
                  <var>.u.</var>
                ex corporum omogenei-
                  <lb/>
                tate, & ex æqualitate magnitudinum corporum antecedentium & conſequentium.</s>
              </p>
              <div xml:id="echoid-div240" type="float" level="4" n="1">
                <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a">
                  <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div242" type="math:theorem" level="3" n="127">
              <head xml:id="echoid-head145" xml:space="preserve">THEOREMA
                <num value="127">CXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1116" xml:space="preserve">SCire etiam nos oportet, quòd quotieſcumque fuerint duo corpora aquea, quo-
                  <lb/>
                rum vnum æqualis magnitudinis ſit alicui miſto, quod quidem miſtum graue
                  <lb/>
                ſit tam in aere, quàm in aqua, alterum verò corpus aquem æqualis ſit magnitudi-
                  <lb/>
                nis alicui corpoli ſimplici, quod quidem corpus ſimplex æqualis ponderis ſit dicto
                  <lb/>
                corpori miſto. </s>
                <s xml:id="echoid-s1117" xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni
                  <lb/>
                tudini corporis miſti, ad pondus corporis aquei, cuius magnitudo æqualis eſt ma-
                  <lb/>
                gnitudini corporis ſimplicis, eadem erit, quæ proportio ponderis alicuius corpo-
                  <lb/>
                ris ſimplicis, cuius magnitudo æqualis ſit magnitudini corporis miſti ſuperius dicti,
                  <lb/>
                ſed ſpeciei corporis ſimplicis iam dicti, ad pondus dicti miſti.</s>
              </p>
              <p>
                <s xml:id="echoid-s1118" xml:space="preserve">Exempli gratia, ſit corpus aqueum
                  <var>.e.</var>
                magnitudinis æqualis corpori
                  <var>.m.</var>
                mixto,
                  <lb/>
                corpus verò aqueum
                  <var>.i.</var>
                æqualis magnitudinis ſit corpori ſimplici
                  <var>.a.</var>
                quod quidem
                  <lb/>
                corpus
                  <var>.a.</var>
                æqualis ponderis ſit cum corpore
                  <var>.m.</var>
                & corpus
                  <var>.u.</var>
                ſit æqualis magnitudinis
                  <lb/>
                cum corpore
                  <var>.m.</var>
                ſed ſpeciei corporis
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s1119" xml:space="preserve">Tunc dico proportionem ponderis
                  <var>.e.</var>
                ad
                  <lb/>
                pondus
                  <var>.i.</var>
                  <reg norm="eandem" type="context">eãdem</reg>
                eſſe, quæ ponderis
                  <var>.u.</var>
                ad pondus
                  <var>.m.</var>
                primum nulli dubium eſt, quin
                  <lb/>
                eadem proportio ſit magnitudinis
                  <var>.e.</var>
                ad magnitudinem
                  <var>.i.</var>
                quæ magnitudinis
                  <var>.m.</var>
                ad
                  <lb/>
                a. ſed
                  <var>.m.</var>
                ad
                  <var>.a.</var>
                eſt vt
                  <var>.u.</var>
                ad
                  <var>.a.</var>
                ex .7. quinti </s>
                <s xml:id="echoid-s1120" xml:space="preserve">quare
                  <lb/>
                ex .11. eiuſdem proportio
                  <var>.e.</var>
                ad
                  <var>.i.</var>
                erit vt
                  <var>.u.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0097-02a" xlink:href="fig-0097-02"/>
                de ipſius magnitudinibus loquendo, ſed propor-
                  <lb/>
                tio ponderis
                  <var>.u.</var>
                ad pondus
                  <var>.a.</var>
                eadem eſt, quæ ma
                  <lb/>
                gnitudinis
                  <var>.u.</var>
                ad magnitudinem
                  <var>.a.</var>
                ex omogenei-
                  <lb/>
                tate. </s>
                <s xml:id="echoid-s1121" xml:space="preserve">Idem dico de pondere
                  <var>.e.</var>
                ad pondus .1. </s>
                <s xml:id="echoid-s1122" xml:space="preserve">Qua-
                  <lb/>
                re proportio ponderis
                  <var>.e.</var>
                ad pondus
                  <var>.i.</var>
                eadem erit
                  <lb/>
                quæ ponderis
                  <var>.u.</var>
                ad pondus
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s1123" xml:space="preserve">Sed ponderis
                  <var>.u.</var>
                  <lb/>
                ad pondus
                  <var>.m.</var>
                eadem eſt quæ ponderis
                  <var>.u.</var>
                ad pondus
                  <var>.a.</var>
                ex .7. quinti, ergò ex .11.
                  <lb/>
                eiuſdem proportio ponderis
                  <var>.e.</var>
                ad pondus
                  <var>.i.</var>
                eadem erit, quæ ponderis
                  <var>.u.</var>
                ad pon-
                  <lb/>
                dus
                  <var>.m.</var>
                quod eſt propoſitum.</s>
              </p>
              <div xml:id="echoid-div242" type="float" level="4" n="1">
                <figure xlink:label="fig-0097-02" xlink:href="fig-0097-02a">
                  <image file="0097-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0097-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div244" type="math:theorem" level="3" n="128">
              <head xml:id="echoid-head146" xml:space="preserve">THEOREMA
                <num value="128">CXXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1124" xml:space="preserve">NVNC ad cognoſcendam proportionem duarum diuerſarum ſpecierum in
                  <lb/>
                corpore miſto propoſito, tribus corporibus aqueis mediantibus, quæ
                  <reg norm="quidem" type="context">quidẽ</reg>
                  <lb/>
                corpora æqualium magnitudinum ſint alijs tribus corporibus vnius & eiuſdem pon
                  <lb/>
                deris, quorum vnum ſit mixtum, reliqua verò duo ſimplicia, ſed ſpecierum mixti,
                  <lb/>
                hoc ordine procedemus.</s>
              </p>
            </div>
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