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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            <div xml:id="echoid-div250" type="math:theorem" level="3" n="131">
              <pb o="89" rhead="THEOR. ARITH." n="101" file="0101" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0101"/>
            </div>
            <div xml:id="echoid-div252" type="math:theorem" level="3" n="132">
              <head xml:id="echoid-head150" xml:space="preserve">THEOREMA
                <num value="132">CXXXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1156" xml:space="preserve">SED quia aliquis poſſet in dubium reuocare, an poſſibile ſit inuenire tertium
                  <lb/>
                terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com
                  <lb/>
                municantibus in tali proportionalitate, hoc eſt harmonica. </s>
                <s xml:id="echoid-s1157" xml:space="preserve">Vthoc oſtendatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1158" xml:space="preserve">Sint duo termini dati
                  <var>.a.o.</var>
                et
                  <var>.a.e.</var>
                inter ſe communicantes, tertius verò inuentus
                  <lb/>
                ſit
                  <var>.a.c.</var>
                qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem
                  <lb/>
                eſſe cum primis datis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1159" xml:space="preserve">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-
                  <lb/>
                dem proportionem eſſe
                  <var>.a.c.</var>
                ad
                  <var>.a.o.</var>
                quæ eſt
                  <var>.e.c.</var>
                ad
                  <var>.e.o.</var>
                vnde permutando ita erit
                  <var>.a.
                    <lb/>
                  c.</var>
                ad
                  <var>.e.c.</var>
                vt
                  <var>.a.o.</var>
                ad
                  <var>.o.e.</var>
                & quia ex .9. decimi Euclid
                  <var>.a.o.</var>
                communicat cum
                  <var>.o.e.</var>
                </s>
                <s xml:id="echoid-s1160" xml:space="preserve">quare
                  <lb/>
                ex .10. eiuſdem
                  <var>.a.c.</var>
                communicabit cum
                  <var>.e.c.</var>
                & per .9. cum
                  <var>.a.e.</var>
                et per .8. cum
                  <var>.a.o.</var>
                  <lb/>
                quod
                  <unsure/>
                eſt propoſitum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1161" xml:space="preserve">Sed ſi datus fuerit maximus
                  <var>.a.c.</var>
                cum medio
                  <var>.a.e.</var>
                interſe communicantes mini-
                  <lb/>
                mum verò
                  <var>.a.o.</var>
                probabo
                  <reg norm="communicantem" type="context">cõmunicantem</reg>
                cum illis eſſe. </s>
                <s xml:id="echoid-s1162" xml:space="preserve">Cogitemus ergo
                  <var>.c.f.</var>
                æqua-
                  <lb/>
                jem eſſe differentiæ
                  <var>.c.e.</var>
                cognitæ, vnde habebimus proportionem,
                  <var>a.c.</var>
                ad
                  <var>.c.f.</var>
                vt
                  <var>.a.o.</var>
                  <lb/>
                ad
                  <var>.o.e.</var>
                & componendo
                  <var>.a.f.</var>
                ad
                  <var>.f.c.</var>
                vt
                  <var>.a.e.</var>
                ad
                  <var>.e.o.</var>
                & quia (ex ſuppoſito).
                  <var>a.c.</var>
                commu-
                  <lb/>
                nicat cum
                  <var>.e.c.</var>
                hoc eſt cum
                  <var>.c.f.</var>
                </s>
                <s xml:id="echoid-s1163" xml:space="preserve">quare
                  <lb/>
                ex eadem .9. dicti decimi
                  <var>.a.f.</var>
                et
                  <var>.f.c.</var>
                  <reg norm="erunt" type="context">erũt</reg>
                  <lb/>
                  <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a" number="138">
                    <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-01"/>
                  </figure>
                inter ſe communicantes, & per .10.
                  <var>a.e.</var>
                  <lb/>
                communicabit cum
                  <var>.o.e.</var>
                & per .9.
                  <var>a.e.</var>
                cò
                  <lb/>
                municabit cum
                  <var>.a.o.</var>
                vnde per .8.
                  <var>a.o.</var>
                communicabit cum
                  <var>.a.c.</var>
                ſimiliter.</s>
              </p>
            </div>
            <div xml:id="echoid-div254" type="math:theorem" level="3" n="133">
              <head xml:id="echoid-head151" xml:space="preserve">THEOREMA
                <num value="133">CXXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1164" xml:space="preserve">SED ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide
                  <lb/>
                remus in dicta proportionalitate, ita faciendum erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1165" xml:space="preserve">Sint, exempli gratia, duo termini dati
                  <var>.q.b.</var>
                et
                  <var>.b.r.</var>
                minor
                  <var>.b.r.</var>
                ex maiori
                  <var>.b.q.</var>
                de-
                  <lb/>
                trahatur, reſiduum verò
                  <var>.q.x.</var>
                multiplicetur per
                  <var>.b.r.</var>
                productum poſteà diuidatur per
                  <lb/>
                  <var>q.r.</var>
                vnde proueniet nobis
                  <var>.x.l.</var>
                pro differentia minori, quæ addita cum
                  <var>.b.x.</var>
                minimo
                  <lb/>
                termino, dabit nobis
                  <var>.b.l.</var>
                mcdium terminum harmonicum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1166" xml:space="preserve">Pro cuius ratione cogitemus dictum medium terminum
                  <var>.b.l.</var>
                iam inuentum eſſe,
                  <lb/>
                vnde ita erit proportio
                  <var>.q.l.</var>
                ad
                  <var>.l.x.</var>
                vt
                  <var>.q.b.</var>
                ad
                  <var>.b.r.</var>
                ex forma huius proportionalitatis,
                  <lb/>
                </s>
                <s xml:id="echoid-s1167" xml:space="preserve">quare coniunctim ita erit
                  <var>.q.r.</var>
                ad
                  <var>.r.b.</var>
                vt
                  <lb/>
                  <var>q.x.</var>
                ad
                  <var>.x.l.</var>
                & proptereà ex .20. ſeptimi
                  <lb/>
                  <figure xlink:label="fig-0101-02" xlink:href="fig-0101-02a" number="139">
                    <image file="0101-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-02"/>
                  </figure>
                productum, quod fit ex
                  <var>.q.r.</var>
                in
                  <var>.x.l.</var>
                æqua-
                  <lb/>
                le erit producto
                  <var>.q.x.</var>
                in
                  <var>.b.r</var>
                . </s>
                <s xml:id="echoid-s1168" xml:space="preserve">Rectè igitur
                  <lb/>
                fit cum diuiditur hoc productum per
                  <var>.q.r.</var>
                vt proueniat nobis
                  <var>.x.l.</var>
                differentia minor.</s>
              </p>
            </div>
            <div xml:id="echoid-div256" type="math:theorem" level="3" n="134">
              <head xml:id="echoid-head152" xml:space="preserve">THEOREMA
                <num value="134">CXXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1169" xml:space="preserve">POſſumus etiam harmonicè diuidere vnam datam proportionem abſque aliqua
                  <lb/>
                diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet.
                  <lb/>
                </s>
                <s xml:id="echoid-s1170" xml:space="preserve">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                inuenian-
                  <lb/>
                tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc
                  <lb/>
                eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-
                  <lb/>
                dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum­ </s>
              </p>
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