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-div247" type="math:theorem" level="3" n="130">
              <p>
                <s xml:id="echoid-s1143" xml:space="preserve">
                  <pb o="88" rhead="IO. BAPT. BENED." n="100" file="0100" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0100"/>
                ducto .2. g. in
                  <var>.d.p.</var>
                ex .20. ſeptimi, proptereà quòd proportio
                  <var>.q.o.</var>
                ad
                  <var>.o.p.</var>
                hoc eſt ad
                  <var>.
                    <lb/>
                  d.p.</var>
                eſt vt
                  <var>.a.g.</var>
                ad
                  <var>.g.n.</var>
                coniunctim cum diſiunctim it a ſit
                  <var>.q.p.</var>
                ad
                  <var>.p.o.</var>
                vt
                  <var>.a.n.</var>
                ad
                  <var>.n.g.</var>
                  <lb/>
                  <reg norm="permutando" type="context">permutãdo</reg>
                eo quòd
                  <var>.q.p.</var>
                ad
                  <var>.a.n.</var>
                (ideſt ad
                  <var>.e.c.</var>
                ) ita ſe
                  <reg norm="hent" type="context">hẽt</reg>
                ut
                  <var>.p.o.</var>
                (hoc eſt
                  <var>.d.p.</var>
                ) ad
                  <var>.n.g.</var>
                  <lb/>
                ex
                  <reg norm="conditionibus" type="context">cõditionibus</reg>
                armonicæ proportio nalitatis. </s>
                <s xml:id="echoid-s1144" xml:space="preserve">Deinde ſi detraxerimus
                  <var>.n.g.</var>
                ex
                  <var>.a.g.</var>
                  <lb/>
                remanebit
                  <var>.e.c.</var>
                minor terminus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1145" xml:space="preserve">Sed ſi
                  <var>.e.c.</var>
                tertius terminus nobis propoſitus eſſet ſimul cum
                  <var>.a.g.</var>
                medio, & volue
                  <lb/>
                rimus maiorem inuenire
                  <var>.q.p.</var>
                ſcilicet, oportebit
                  <var>.e.c.</var>
                ex
                  <var>.a.g.</var>
                detrahere, differentiam
                  <lb/>
                verò
                  <var>.n.g.</var>
                ſimiliter demeremus
                  <lb/>
                ex
                  <var>.e.c.</var>
                unde remaneret nobis
                  <var>.e.t.</var>
                  <lb/>
                  <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a" number="136">
                    <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-01"/>
                  </figure>
                cognitum, quo reſiduo
                  <var>.c.t.</var>
                me-
                  <lb/>
                diante diuidemus productum,
                  <reg norm="quod" type="simple">ꝙ</reg>
                  <lb/>
                furgit ex
                  <var>.a.g.</var>
                in
                  <var>.t.c.</var>
                & prouentus
                  <var>.
                    <lb/>
                  d.p.</var>
                erit differentia maior, eo
                  <reg norm="quod" type="simple">ꝙ</reg>
                  <lb/>
                  <reg norm="productum" type="context">productũ</reg>
                quod ſit ex
                  <var>.e.t.</var>
                in
                  <var>.d.p.</var>
                  <lb/>
                æquale eſt producto quòd fit ex
                  <var>.a.g.</var>
                in
                  <var>.t.c.</var>
                per 20. ſeptimi Eucli. eo quòd
                  <var>.a.g.</var>
                (id-
                  <lb/>
                eſt
                  <var>.q.d.</var>
                ) ad
                  <var>.d.p.</var>
                eſt ut
                  <var>.e.t.</var>
                ad
                  <var>.t.c.</var>
                diſiunctim, cum coniunctim ita ſit
                  <var>.q.p.</var>
                ad
                  <var>.d.p.</var>
                vt
                  <var>.e.
                    <lb/>
                  c.</var>
                ad
                  <var>.t.c.</var>
                permutando, quia
                  <var>.q.p.</var>
                ad
                  <var>.e.c.</var>
                eſt vt
                  <var>.d.p.</var>
                ad
                  <var>.t.c.</var>
                hoc eſt ad
                  <var>.n.g.</var>
                ex legibus
                  <lb/>
                dictis.</s>
              </p>
            </div>
            <div xml:id="echoid-div250" type="math:theorem" level="3" n="131">
              <head xml:id="echoid-head149" xml:space="preserve">THEOREMA
                <num value="131">CXXXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s1146" xml:space="preserve">ALIA etiam methodo hoc perfici poſſe comperi. </s>
                <s xml:id="echoid-s1147" xml:space="preserve">Propoſiti enim cum nobis fue
                  <lb/>
                rint duo termini
                  <var>.c.e.</var>
                minimus et
                  <var>.g.a.</var>
                medius, maximus verò quærendus ſit, de
                  <lb/>
                trahatur differentia
                  <var>.g.n.</var>
                ex
                  <var>.e.c.</var>
                & per reſiduum
                  <var>.e.t.</var>
                diuidatur productum
                  <reg norm="quod" type="simple">ꝙ</reg>
                fit ex
                  <var>.a.
                    <lb/>
                  g.</var>
                in
                  <var>.e.c.</var>
                prouentus quæ erit
                  <var>.q.p.</var>
                terminus quæſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1148" xml:space="preserve">Pro cuius ratione, ponamus in eſſe terminum
                  <var>.q.p.</var>
                </s>
                <s xml:id="echoid-s1149" xml:space="preserve">tunc ex forma huius proportio
                  <lb/>
                nalitatis nulli dubium erit quin
                  <var>.q.p.</var>
                ad
                  <var>.e.c.</var>
                fit vt
                  <var>.d.p.</var>
                ad
                  <var>.n.g.</var>
                hoc eft ad
                  <var>.t.c.</var>
                vnde ex
                  <lb/>
                19. quinti vel .12. ſeptimi ita eſſet
                  <var>.q.d.</var>
                ad
                  <var>.e.t.</var>
                vt
                  <var>.q.p.</var>
                ad
                  <var>.e.c.</var>
                </s>
                <s xml:id="echoid-s1150" xml:space="preserve">quare ex .20. @cptimi pro
                  <lb/>
                ductum
                  <reg norm="quod" type="simple">ꝙ</reg>
                naſcitur ex
                  <var>.p.d.</var>
                (hoc eſt
                  <var>.a.g.</var>
                ) in
                  <var>.e.c.</var>
                æquale eric producto
                  <var>.e.t.</var>
                in
                  <var>.q.p.</var>
                qua-
                  <lb/>
                propter ſi diuiſerimus id per
                  <var>.e.t.</var>
                proueniet nobis
                  <var>.q.p</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1151" xml:space="preserve">Sed
                  <reg norm="cum" type="context">cũ</reg>
                nobis propoſiti fuerint duo termini
                  <var>.q.p.</var>
                maximus, et
                  <var>.a.g.</var>
                medius, ſi
                  <reg norm="mini- mum" type="context">mini-
                    <lb/>
                  mũ</reg>
                  <var>.e.c.</var>
                  <reg norm="voluerimus" type="simple">voluerimꝰ</reg>
                inuenire. </s>
                <s xml:id="echoid-s1152" xml:space="preserve">Termino
                  <var>.q.p.</var>
                maximo,
                  <reg norm="iungatur" type="context simple">iũgat̃</reg>
                .
                  <var>p.o.</var>
                ęqualis,
                  <var>p.d.</var>
                  <reg norm="differentię" type="context">differẽtię</reg>
                  <lb/>
                propoſitæ, diuidatur poſtea productum
                  <reg norm="quod" type="simple">ꝙ</reg>
                ex
                  <var>.q.p.</var>
                in
                  <var>.a.g.</var>
                generatur per
                  <var>.q.o.</var>
                prouen
                  <lb/>
                tus autem ſit
                  <var>.e.c.</var>
                qui quidem erit terminus quæſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1153" xml:space="preserve">Cuius operationis ſpeculutio hæc erit, ſupponatur terminum
                  <var>.e.c.</var>
                inuentum eſſe
                  <lb/>
                vnde
                  <var>.n.g.</var>
                differentia ſit inter
                  <var>.e.c.</var>
                  <lb/>
                et
                  <var>.a.g.</var>
                ex forma igitur armonicæ
                  <lb/>
                  <figure xlink:label="fig-0100-02" xlink:href="fig-0100-02a" number="137">
                    <image file="0100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-02"/>
                  </figure>
                proportionalitis ita erit
                  <var>.q.p.</var>
                ad
                  <var>.a.
                    <lb/>
                  n.</var>
                vt
                  <var>.p.o.</var>
                ad
                  <var>.n.g.</var>
                vnde ex .13. quin-
                  <lb/>
                ti. </s>
                <s xml:id="echoid-s1154" xml:space="preserve">Ita erit
                  <var>.q.o.</var>
                ad
                  <var>.a.g.</var>
                vt
                  <var>.q.p.</var>
                ad
                  <var>.a.
                    <lb/>
                  n.</var>
                ergo
                  <reg norm="productum" type="context">productũ</reg>
                quòd fit ex
                  <var>.a.g.</var>
                  <lb/>
                in
                  <var>.q.p.</var>
                (ex .20. ſeptimi) æquale erit
                  <lb/>
                producto
                  <var>.q.o.</var>
                in
                  <var>.a.n</var>
                . </s>
                <s xml:id="echoid-s1155" xml:space="preserve">Quare ſi diuiſum fuerit tale productum per
                  <var>.q.o.</var>
                proueniet no-
                  <lb/>
                bis
                  <var>.e.c.</var>
                quòd querebamus.</s>
              </p>
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