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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IO. BAPT. BENED.
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          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div256" type="math:theorem" level="3" n="134">
              <p>
                <s xml:id="echoid-s1170" xml:space="preserve">
                  <pb o="90" rhead="IO. BAPT. BENED." n="102" file="0102" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0102"/>
                mæ iam dictæ in maiorem eorum, hoc eſt quod fit ex quinque in .3. quod erit .15. </s>
                <s xml:id="echoid-s1171" xml:space="preserve">Vt
                  <lb/>
                autem medium terminum harmonicum inter iſtos habeamus, accipiatur
                  <reg norm="duplum" type="context">duplũ</reg>
                pro-
                  <lb/>
                ducti, quod fit ex primis minimis terminis, quod erit .12.</s>
              </p>
              <p>
                <s xml:id="echoid-s1172" xml:space="preserve">Cuius rei ſpeculatio eſt iſta: </s>
                <s xml:id="echoid-s1173" xml:space="preserve">ſignificentur duo termini datæ proportionis ab
                  <var>.q.b.</var>
                  <lb/>
                et
                  <var>.b.r.</var>
                quorum ſumma erit
                  <var>.q.r.</var>
                cuius quadratum ſit
                  <var>.q.o.</var>
                ſit etiam imaginata
                  <var>.b.e.</var>
                  <lb/>
                parallela ad
                  <var>.o.r</var>
                . </s>
                <s xml:id="echoid-s1174" xml:space="preserve">
                  <reg norm="Sitque" type="simple">Sitq́;</reg>
                  <var>.b.x.</var>
                æqualis
                  <var>.b.r.</var>
                et
                  <var>.q.u.</var>
                ſimiliter, & ducatur
                  <var>.x.y.</var>
                parallela ad
                  <lb/>
                  <var>r.o.</var>
                et
                  <var>.u.l.</var>
                ad
                  <var>.q.x</var>
                . </s>
                <s xml:id="echoid-s1175" xml:space="preserve">Tunc habebimus
                  <var>.b.o.</var>
                æquale ei producto, quod fit ex
                  <var>.q.r.</var>
                in
                  <var>.b.r.</var>
                  <lb/>
                et
                  <var>.b.y.</var>
                eidem etiam æquale, et
                  <var>.q.e.</var>
                pro producto, quod fit ex
                  <var>.q.r.</var>
                in
                  <var>.q.b.</var>
                et
                  <var>.q.l.</var>
                pro
                  <lb/>
                eo, quod fit ex
                  <var>.q.x.</var>
                in
                  <var>.b.r</var>
                . </s>
                <s xml:id="echoid-s1176" xml:space="preserve">Vnde
                  <var>.q.l.</var>
                cum
                  <var>.b.y.</var>
                æquale fiet duplo ei, quod fit ex
                  <var>.q.b.</var>
                  <lb/>
                in
                  <var>.b.r</var>
                . </s>
                <s xml:id="echoid-s1177" xml:space="preserve">Dico nunc
                  <var>.b.o.</var>
                eſſe minimum terminum eorum, quos quærimus, et
                  <var>.y.b.</var>
                cum
                  <var>.
                    <lb/>
                  x.u.</var>
                medium
                  <var>.q.e.</var>
                verò maximum huiuſmodi proportionalitatis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1178" xml:space="preserve">Primum ergo certi ſcimus ex prima ſexti vel .18. ſeptimi eandem exiſtere pro-
                  <lb/>
                portionem
                  <var>.q.e.</var>
                ad
                  <var>.b.o.</var>
                ſeu ad
                  <var>.b.y.</var>
                quæ
                  <var>.q.b.</var>
                ad
                  <var>.b.r</var>
                : ſed
                  <var>.u.y.</var>
                ad
                  <var>.u.x.</var>
                eſt vt
                  <var>.y.l.</var>
                ad
                  <var>.l.x.</var>
                  <lb/>
                hoc eſt vt
                  <var>.q.b.</var>
                ad
                  <var>.b.r.</var>
                ideſt vt
                  <var>.q.e.</var>
                ad
                  <var>.b.o.</var>
                & ſumma
                  <var>.u.y.</var>
                cum
                  <var>.u.x.</var>
                ideſt
                  <var>.q.y.</var>
                minor eſt
                  <lb/>
                quam
                  <var>.q.e.</var>
                maximus terminus per
                  <var>.b.y.</var>
                minimum ter-
                  <lb/>
                minum. </s>
                <s xml:id="echoid-s1179" xml:space="preserve">&
                  <reg norm="coniunctim" type="context">cõiunctim</reg>
                  <var>.q.y.</var>
                ad
                  <var>.q.l.</var>
                vt
                  <var>.y.x.</var>
                ad
                  <var>.x.l.</var>
                hoc eſt
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/>
                vt
                  <var>.q.r.</var>
                ad
                  <var>.r.b</var>
                . </s>
                <s xml:id="echoid-s1180" xml:space="preserve">Vnde ex ſpeculatione
                  <reg norm="præcedentis" type="context">præcedẽtis</reg>
                theo
                  <lb/>
                rematis, ſequitur
                  <var>.u.y.</var>
                eſſe differentiam inter
                  <reg norm="maximum" type="context">maximũ</reg>
                  <lb/>
                & medium terminum, et
                  <var>.u.x.</var>
                eſſe differentiam inter
                  <lb/>
                medium & minimum dictæ proportionalitatis. </s>
                <s xml:id="echoid-s1181" xml:space="preserve">Nam
                  <lb/>
                eadem proportio eſt
                  <var>.q.e.</var>
                maximi termini ad
                  <var>.b.o.</var>
                mi-
                  <lb/>
                nimi. quæ
                  <var>.u.y.</var>
                (differentia inter
                  <var>.q.e.</var>
                & gnomonem
                  <var>.
                    <lb/>
                  u.b.y.</var>
                ) ad
                  <var>.u.x.</var>
                (differentia inter dictum
                  <var>.u.b.y.</var>
                et
                  <var>.b.y.</var>
                  <lb/>
                minimum terminum, quia ſunt ambæ ut
                  <var>.q.b.</var>
                ad
                  <var>.b.r.</var>
                  <lb/>
                vt diximus. </s>
                <s xml:id="echoid-s1182" xml:space="preserve">Quare
                  <var>.b.y.</var>
                  <reg norm="coniunctum" type="context">coniunctũ</reg>
                cum
                  <var>.x.u.</var>
                medius
                  <lb/>
                terminus erit, qui quidem (vt dictum eſt) duplus eſt ei
                  <lb/>
                quod fit ex
                  <var>.q.b.</var>
                in
                  <var>.b.r</var>
                .</s>
              </p>
              <div xml:id="echoid-div256" type="float" level="4" n="1">
                <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a">
                  <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div258" type="math:theorem" level="3" n="135">
              <head xml:id="echoid-head153" xml:space="preserve">THEOREMA
                <num value="135">CXXXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s1183" xml:space="preserve">ALIVM etiam modum ab antiquis traditum ad hoc problema perficiendum
                  <lb/>
                inueni, qui talis eſt. </s>
                <s xml:id="echoid-s1184" xml:space="preserve">Inueniatur primo inter datos terminos extremos, me-
                  <lb/>
                dius terminus in arithmetica proportione, per
                  <reg norm="quem" type="context">quẽ</reg>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0102-02a" xlink:href="fig-0102-02"/>
                multiplicetur vnuſquiſque dictorum extremorum,
                  <lb/>
                deinde multiplicentur ipſi extremi interſe, vnde
                  <lb/>
                habebimus tria producta eadem proportione inui
                  <lb/>
                cem exiſtentia, vt quærebatur.</s>
              </p>
              <div xml:id="echoid-div258" type="float" level="4" n="1">
                <figure xlink:label="fig-0102-02" xlink:href="fig-0102-02a">
                  <image file="0102-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0102-02"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s1185" xml:space="preserve">Exempli gratia, ponamus duos propoſitos ter-
                  <lb/>
                minos eſſe .3. et .2. quorum medius arithmeticè
                  <lb/>
                eſſet .2. cum dimidia vnitate, per quem cum vnum
                  <lb/>
                quemque priorum multiplicauerimus,
                  <reg norm="emergent" type="context">emergẽt</reg>
                no-
                  <lb/>
                bis duo producta, quorum primum ideſt maius eſſet
                  <lb/>
                7. cum dimidia vnitate, reliquum verò eſſet
                  <lb/>
                quinque, productum poſteà quod ex ipſis extremis
                  <lb/>
                prouenit, erit .6. quod quidem eſt harmonicè collo
                  <lb/>
                catum inter .7. cum dimidia vnitate, & quinque.</s>
              </p>
              <p>
                <s xml:id="echoid-s1186" xml:space="preserve">Cuius rei ſpeculatio omnis à præcedenti theore-
                  <lb/>
                mate dependet. </s>
                <s xml:id="echoid-s1187" xml:space="preserve">Sint exempli gratia, duo termini </s>
              </p>
            </div>
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