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-div278" type="math:theorem" level="3" n="145">
              <p>
                <s xml:id="echoid-s1275" xml:space="preserve">
                  <reg norm="con- ſequenti" type="context">
                    <pb o="98" rhead="IO. BAPT. BENED." n="110" file="0110" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0110"/>
                  ſequenti</reg>
                reſiduę proportionis; </s>
                <s xml:id="echoid-s1276" xml:space="preserve">quæ quidem reſidua proportio eſſet vt .4. ad .3. hoc
                  <lb/>
                eſt ſeſquitertia, & ſic de cæteris.</s>
              </p>
              <p>
                <s xml:id="echoid-s1277" xml:space="preserve">Pro cuius ratione, ſit proportio
                  <var>.x.</var>
                ad
                  <var>.n.</var>
                ea quæ (exempli gratia) maior ſit, à
                  <lb/>
                qua volumus demere proportionem
                  <var>.t.</var>
                ad
                  <var>.u.</var>
                minorem ſcilicet. </s>
                <s xml:id="echoid-s1278" xml:space="preserve">Nunc autem
                  <lb/>
                productum
                  <var>.x.</var>
                in
                  <var>.u.</var>
                ſit
                  <var>.a.g.</var>
                illud verò
                  <var>.t.</var>
                in
                  <var>.
                    <lb/>
                  n.</var>
                ſit
                  <var>.a.d</var>
                . </s>
                <s xml:id="echoid-s1279" xml:space="preserve">Tunc dico proportionem
                  <var>.a.g.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                  <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a" number="152">
                    <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0110-01"/>
                  </figure>
                d. eſſe reſiduam quæſitam. </s>
                <s xml:id="echoid-s1280" xml:space="preserve">Sit
                  <var>.b.a.</var>
                productum
                  <lb/>
                u. in
                  <var>.n.</var>
                vnde eadem proportio erit producti
                  <var>.a.
                    <lb/>
                  g.</var>
                ad productum
                  <var>.a.b.</var>
                quę
                  <var>.x.</var>
                ad
                  <var>.n.</var>
                et
                  <var>.a.d.</var>
                ad
                  <var>a.b.</var>
                  <lb/>
                quæ
                  <var>.t.</var>
                ad
                  <var>.u.</var>
                ex prima ſexti, ſeu .18. vel .19. ſe-
                  <lb/>
                ptimi, ſed proportio
                  <var>.a.g.</var>
                ad
                  <var>.a.b.</var>
                hoc eſt
                  <var>.x.</var>
                ad
                  <var>.
                    <lb/>
                  n.</var>
                componitur ex ea, quæ eſt
                  <var>.a.g.</var>
                ad
                  <var>.a.d.</var>
                & ea,
                  <lb/>
                quæ eſt
                  <var>.a.d.</var>
                ad
                  <var>.a.b.</var>
                hoc eſt
                  <var>.t.</var>
                ad
                  <var>.u.</var>
                ergò ea, quę
                  <lb/>
                eſt
                  <var>.a.g.</var>
                ad
                  <var>.a.d.</var>
                erit quàm quærebamus.</s>
              </p>
            </div>
            <div xml:id="echoid-div280" type="math:theorem" level="3" n="146">
              <head xml:id="echoid-head165" xml:space="preserve">THEOREMA
                <num value="146">CXLVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s1281" xml:space="preserve">RATIO verò, quòd rectè fiat, quotieſcunque aliquam proportionem dupli-
                  <lb/>
                care volentes, quadramus terminos ipſius proportionis, vel ſi eam triplicare
                  <lb/>
                voluerimus, cubamus ipſos terminos, vel ſi eam quadruplicare voluerimus
                  <lb/>
                inuenimus cenſicos cenſicos terminorum ipſius proportionis, & ſic de ſingulis, in
                  <ref id="ref-0016">.17
                    <lb/>
                  Theo. huiuſmodi tractatus</ref>
                manifeſta eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div281" type="math:theorem" level="3" n="147">
              <head xml:id="echoid-head166" xml:space="preserve">THEOREMA
                <num value="147">CXLVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1282" xml:space="preserve">QVotieſcunque nobis propoſiti fuerint duo numeri ad libitum, deſideraremus­
                  <lb/>
                q́ue duas proportiones tali relatione inuicem refertas, quali ſunt hi duo pro
                  <lb/>
                poſiti numeri inter ſe, ita faciendum erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1283" xml:space="preserve">Sciendum primo eſt proportionem maioris numeri propoſiti ad minorem ſem-
                  <lb/>
                per eſſe alicuius ex quinque generum, hoc eſt aut erit generis multiplicis, aut ſu-
                  <lb/>
                perparticularis, aut multiplicis ſuperparticularis, aut ſuper partientis, aut multi-
                  <lb/>
                plicis ſuperpartientis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1284" xml:space="preserve">Nunc autem ſi erit ex genere multiplici, iam ab antiquis traditus eſt modus,
                  <reg norm="quem" type="context">quẽ</reg>
                  <lb/>
                ſequi debemus. </s>
                <s xml:id="echoid-s1285" xml:space="preserve">Cuius ſpeculatio à me inuenta patet .in .17. Theo. huius libri, vt
                  <lb/>
                in præcedenti dixi.</s>
              </p>
              <p>
                <s xml:id="echoid-s1286" xml:space="preserve">Sed ſi talis proportio datorum numerorum erit alicuius aliorum generum, ita
                  <lb/>
                agemus, ſi fuerit ſuperparticularis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1287" xml:space="preserve">Sit exempli gratia, ſeſquialtera, tunc ſumantur duo numeri inuicem inæquales,
                  <lb/>
                quos à caſu volueris
                  <var>.o.</var>
                et
                  <var>.c.</var>
                qui quidem cubentur, & eorum cubi ſint
                  <var>.a.</var>
                et
                  <var>.e</var>
                . </s>
                <s xml:id="echoid-s1288" xml:space="preserve">Inuenia
                  <lb/>
                tur poſteà. u. ita proportionatus ad
                  <var>.o.</var>
                vt
                  <var>.o.</var>
                eſt ad
                  <var>.c.</var>
                ex regula de tribus, hoc eſt diui-
                  <lb/>
                dendo quadratum ipſius
                  <var>.o.</var>
                per
                  <var>.c.</var>
                vnde nobis proueniat
                  <var>.u.</var>
                & quia proportio
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                  <lb/>
                tripla eſt proportioni
                  <var>.o.</var>
                ad
                  <var>.c.</var>
                & proportio
                  <var>.u.</var>
                ad
                  <var>.c.</var>
                dupla eſt
                  <reg norm="eidem" type="context">eidẽ</reg>
                , quæ
                  <var>.o.</var>
                ad
                  <var>.c.</var>
                ideo
                  <lb/>
                proportio
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                ſeſquialtera erit proportioni
                  <var>.u.</var>
                ad
                  <var>.c</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1289" xml:space="preserve">Sed ſi proportio numerorum propoſitorum fuerit ſeſquitertia, faciemus
                  <var>.a.</var>
                et
                  <var>.e.</var>
                  <lb/>
                eſſe cenſica cenſica ipſius
                  <var>.o.</var>
                et
                  <var>.c</var>
                . </s>
                <s xml:id="echoid-s1290" xml:space="preserve">tunc ſumemus
                  <var>.u.</var>
                conſequentem ad
                  <var>.o.</var>
                vt dictum eſt,
                  <lb/>
                deinde inueniremus
                  <var>.i.</var>
                conſequens ad
                  <var>.u.</var>
                ita ut
                  <var>.u.</var>
                conſequens ipſius
                  <var>.o</var>
                . </s>
                <s xml:id="echoid-s1291" xml:space="preserve">tunc habebi-
                  <lb/>
                mus proportionem
                  <var>.i.</var>
                ad
                  <var>.c.</var>
                triplam, & eam quæ eſt
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                quadruplam proportio- </s>
              </p>
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