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-div163" type="math:theorem" level="3" n="83">
              <p>
                <s xml:id="echoid-s729" xml:space="preserve">
                  <pb o="55" rhead="THEOR. ARITH." n="67" file="0067" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0067"/>
                æqualis .e: et
                  <var>.q.n.</var>
                æqualis
                  <var>.i</var>
                . </s>
                <s xml:id="echoid-s730" xml:space="preserve">Nunc co-
                  <lb/>
                  <figure xlink:label="fig-0067-01" xlink:href="fig-0067-01a" number="92">
                    <image file="0067-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-01"/>
                  </figure>
                gitemus abſolui corpus
                  <var>.n.h.</var>
                ita ut
                  <var>.b.
                    <lb/>
                  o.c.</var>
                ſit vnica recta linea, ex quo ex .25.
                  <lb/>
                vndecimi proportio
                  <var>.n.h.</var>
                ad
                  <var>.n.k.</var>
                ea-
                  <lb/>
                dem eſt quæ
                  <var>.o.h.</var>
                ad
                  <var>o.k.</var>
                ſed ſic ſe ha-
                  <lb/>
                bet
                  <var>.o.h.</var>
                ad
                  <var>.o.k.</var>
                vt
                  <var>.h.b.</var>
                ad
                  <var>.b.k.</var>
                  <lb/>
                ex prima ſexti aut .18. vel .19. ſe-
                  <lb/>
                ptimi itaque
                  <var>.n.h.</var>
                ad
                  <var>.n.k.</var>
                ex .11.
                  <lb/>
                quinti ſic ſe habebit. vt
                  <var>.h.b.</var>
                ad
                  <var>.b.k.</var>
                  <lb/>
                ſed
                  <var>.n.h.</var>
                ad
                  <var>.n.d.</var>
                ex eiſdem ſic ſe habet
                  <lb/>
                ut
                  <var>.h.u.</var>
                ad
                  <var>.d.u.</var>
                et
                  <var>.h.u.</var>
                ad
                  <var>.u.d.</var>
                ita ut
                  <var>.h.
                    <lb/>
                  b.</var>
                ad
                  <var>.b.k.</var>
                ex præſuppoſito. </s>
                <s xml:id="echoid-s731" xml:space="preserve">Itaque ex
                  <lb/>
                11. prædicta
                  <var>.n.h.</var>
                ad
                  <var>.n.k.</var>
                eadem erit
                  <lb/>
                proportio quæ
                  <var>.n.h.</var>
                ad
                  <var>.n.d</var>
                . </s>
                <s xml:id="echoid-s732" xml:space="preserve">Quare
                  <lb/>
                ex .9. quinti
                  <var>.n.k.</var>
                æqualis erit
                  <var>.n.d.</var>
                  <lb/>
                Quod erat propoſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div165" type="math:theorem" level="3" n="84">
              <head xml:id="echoid-head101" xml:space="preserve">THEOREMA
                <num value="84">LXXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s733" xml:space="preserve">CVR quadrato vnius quantitatis radice proportionalis, per ſingulos tres termi
                  <lb/>
                nos diuiſo, prouenientia, ſingulis dictis terminis ſint æqualia.</s>
              </p>
              <p>
                <s xml:id="echoid-s734" xml:space="preserve">
                  <reg norm="Exempli" type="context">Exẽpli</reg>
                gratia, datis tribus terminis continuis proportionalibus .9. 6. 4. qua
                  <lb/>
                dratum medij erit .36. quod per .9. diuiſum dabit .4: per .6: 6. per .4: 9.</s>
              </p>
              <p>
                <s xml:id="echoid-s735" xml:space="preserve">Cuius gratia, ſint tres termini
                  <reg norm="continui" type="context">cõtinui</reg>
                  <reg norm="proportionales" type="simple">ꝓportionales</reg>
                  <var>.a.o</var>
                :
                  <var>o.c.</var>
                et
                  <var>.c.q.</var>
                  <reg norm="quadratum" type="context">quadratũ</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                  <lb/>
                medij ſit
                  <var>.e.c</var>
                . </s>
                <s xml:id="echoid-s736" xml:space="preserve">Iam ſi applicetur
                  <reg norm="rectangulum" type="context">rectangulũ</reg>
                  <var>.a.d.</var>
                æquale quadrato
                  <var>.e.c.</var>
                ipſi
                  <var>.a.o.</var>
                & re-
                  <lb/>
                ctangulum
                  <var>.q.p.</var>
                æquale eidem quadrato
                  <var>.e.c.</var>
                ipſi
                  <var>.c.q.</var>
                ſi quadratum
                  <var>.e.c.</var>
                per
                  <var>.a.o.</var>
                diui
                  <lb/>
                ſerimus, proueniens erit
                  <var>.o.d.</var>
                  <reg norm="diuiſoque" type="simple">diuiſoq́</reg>
                per
                  <var>.c.q.</var>
                proueniens erit
                  <var>.c.p.</var>
                quod ſi per ſuam
                  <lb/>
                radicem
                  <var>.o.c.</var>
                diuidatur, proueniens erit
                  <var>.o.</var>
                  <lb/>
                  <figure xlink:label="fig-0067-02" xlink:href="fig-0067-02a" number="93">
                    <image file="0067-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0067-02"/>
                  </figure>
                e. quod ſine dubio æquale eſt
                  <var>.o.c.</var>
                ſed dico
                  <var>.
                    <lb/>
                  o.d.</var>
                æqualem eſſe
                  <var>.c.q</var>
                . </s>
                <s xml:id="echoid-s737" xml:space="preserve">Nam ex .16. ſexti aut
                  <lb/>
                20. ſeptimi eadem eſt proportio
                  <var>.a.o.</var>
                ad
                  <var>.o.
                    <lb/>
                  c.</var>
                quę
                  <var>.o.e.</var>
                ad
                  <var>.o.d.</var>
                nempe
                  <var>.o.c.</var>
                ad
                  <var>.o.d.</var>
                itaque
                  <lb/>
                  <var>o.d.</var>
                ex .9. quinti æqualis eſt
                  <var>.c.q.</var>
                quandoqui
                  <lb/>
                dem ex .11. ſic ſe habet
                  <var>.o.c.</var>
                ad
                  <var>.o.d.</var>
                ſicut
                  <var>.o.
                    <lb/>
                  c.</var>
                ad
                  <var>.c.q</var>
                . </s>
                <s xml:id="echoid-s738" xml:space="preserve">Applicatis ijſdem rationibus ipſi
                  <var>.
                    <lb/>
                  p.c.</var>
                probabimus
                  <var>.c.p.</var>
                æqualem eſſe
                  <var>.a.o.</var>
                cum
                  <lb/>
                  <var>o.c.</var>
                media ſit proportionalis,
                  <reg norm="tam" type="context">tã</reg>
                inter
                  <var>.c.p.</var>
                et
                  <lb/>
                  <var>c.q.</var>
                quam inter
                  <var>.a.o.</var>
                et
                  <var>.c.q.</var>
                itaque
                  <var>.c.p.</var>
                æqua-
                  <lb/>
                lis eſt
                  <var>.a.o</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div167" type="math:theorem" level="3" n="85">
              <head xml:id="echoid-head102" xml:space="preserve">THEOREMA
                <num value="85">LXXXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s739" xml:space="preserve">CVR propoſitis tribus quantitatibus continuis proportionalibus proportione
                  <lb/>
                aliarum duarum nobis datarum, multiplicata maiori poſtremarum dua-
                  <lb/>
                rum in ſummam mediæ cum minima trium primarum, productum æqua-
                  <lb/>
                le ſit producto minoris duarum in ſummam maximæ cum media trium.</s>
              </p>
              <p>
                <s xml:id="echoid-s740" xml:space="preserve">Exempli gratia proponuntur quantitates .9. 6. 4. proportione numerorum pro- </s>
              </p>
            </div>
          </div>
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