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-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div167" type="math:theorem" level="3" n="85">
              <p>
                <s xml:id="echoid-s740" xml:space="preserve">
                  <pb o="56" rhead="IO. BAPT. BENED." n="68" file="0068" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0068"/>
                poſitorum .3. et .2. multiplicato .3. per .10.
                  <reg norm="ſummam" type="context">ſummã</reg>
                .6. cum .4. dantur .30. quod pro-
                  <lb/>
                ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</s>
              </p>
              <p>
                <s xml:id="echoid-s741" xml:space="preserve">Quod vt cognoſcamus, tres quan
                  <lb/>
                  <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a" number="94">
                    <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-01"/>
                  </figure>
                titates continuæ proportionales ſint
                  <lb/>
                  <var>b.a.p.</var>
                proportione
                  <var>.d.q.</var>
                productum
                  <lb/>
                autem
                  <var>.d.</var>
                in ſummam
                  <var>.a.</var>
                cum
                  <var>.p.</var>
                ſit
                  <var>.f.t.</var>
                  <lb/>
                & productum
                  <var>.q.</var>
                in ſummam
                  <var>.b.a.</var>
                ſit
                  <var>.
                    <lb/>
                  K.h.</var>
                et
                  <var>.K.n.</var>
                ſit æqualis
                  <var>.b.</var>
                et
                  <var>.n.o.</var>
                æqua
                  <lb/>
                lis
                  <var>.a.</var>
                & ita etiam
                  <var>.o.u.</var>
                eidem
                  <var>.a.</var>
                et
                  <var>.u.t.</var>
                  <lb/>
                æqualis
                  <var>.p.</var>
                et
                  <var>.h.o.</var>
                ipſi
                  <var>.q.</var>
                et
                  <var>.f.o.</var>
                ipſi
                  <var>.d</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s742" xml:space="preserve">quare ita ſe habebit
                  <var>.K.n.</var>
                ad
                  <var>.n.o.</var>
                ſicut
                  <lb/>
                  <var>o.u.</var>
                ad
                  <var>.u.t.</var>
                & componendo
                  <var>.K.o.</var>
                ad
                  <var>.
                    <lb/>
                  n.o.</var>
                vt
                  <var>.o.t.</var>
                ad
                  <var>.u.t.</var>
                & permutando
                  <var>.K.
                    <lb/>
                  o.</var>
                ad
                  <var>.o.t.</var>
                vt
                  <var>.n.o.</var>
                hoc eſt
                  <var>.o.u.</var>
                ad
                  <var>.u.t.</var>
                &
                  <lb/>
                pariter
                  <var>.f.o.</var>
                ad
                  <var>.o.h.</var>
                vt
                  <var>.o.u.</var>
                ad
                  <var>.u.t</var>
                . </s>
                <s xml:id="echoid-s743" xml:space="preserve">Ita-
                  <lb/>
                que ſicut
                  <var>.k.o.</var>
                ad
                  <var>.o.t.</var>
                ex quo ex .15. ſexti aut .20. ſeptimi
                  <var>.K.h.</var>
                æqualis erit
                  <var>.f.t</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div169" type="math:theorem" level="3" n="86">
              <head xml:id="echoid-head103" xml:space="preserve">THEOREMA
                <num value="86">LXXXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s744" xml:space="preserve">CVR multiplicatis ſingulis tribus quantitatibus continuis proportionalibus in
                  <lb/>
                reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum trium
                  <lb/>
                in mediam proportionalem.</s>
              </p>
              <p>
                <s xml:id="echoid-s745" xml:space="preserve">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro
                  <lb/>
                ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et
                  <num value="4">.
                    <lb/>
                  4.</num>
                in .6: 24. quæ producta ſimul collecta efficiunt numerum .228 ſed
                  <reg norm="tantum" type="context">tantũ</reg>
                eſt pro-
                  <lb/>
                ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</s>
              </p>
              <p>
                <s xml:id="echoid-s746" xml:space="preserve">Cuius
                  <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg>
                cauſa, tres termini
                  <reg norm="continui" type="context">cõtinui</reg>
                proportionales ſignificentur linea
                  <var>.
                    <lb/>
                  b.e.</var>
                nempe
                  <var>.b.d</var>
                :
                  <var>d.c</var>
                :
                  <var>c.e.</var>
                cuius duplum ſit
                  <var>.u.e.</var>
                et
                  <var>.b.f.</var>
                æqualis ſit
                  <var>.b.d.</var>
                et
                  <var>.f.n</var>
                :
                  <var>d.c.</var>
                et
                  <var>.n.u</var>
                :
                  <lb/>
                c. e productum verò
                  <var>.u.e.</var>
                in
                  <var>.d.c.</var>
                ſit
                  <var>.u.s.</var>
                cui dico æqualem eſſe ſummam productorum
                  <lb/>
                ſingulorum trium terminorum in reliquos duos. </s>
                <s xml:id="echoid-s747" xml:space="preserve">Quamobrem ducantur perpendi-
                  <lb/>
                culares
                  <var>.c.g</var>
                :
                  <var>d.o</var>
                :
                  <var>b.i</var>
                :
                  <var>f.a.</var>
                et
                  <var>.n.p.</var>
                inter
                  <var>.u.e.</var>
                et
                  <var>.q.s.</var>
                ex quo pro producto
                  <var>.c.e.</var>
                in
                  <var>.c.d.</var>
                ha-
                  <lb/>
                bebimus rectangulum
                  <var>.c.s.</var>
                & rectan-
                  <lb/>
                  <figure xlink:label="fig-0068-02" xlink:href="fig-0068-02a" number="95">
                    <image file="0068-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-02"/>
                  </figure>
                gulum
                  <var>.d.g.</var>
                pro producto
                  <var>.c.e.</var>
                in
                  <var>.d.b.</var>
                  <lb/>
                ex .16. ſexti aut .20. ſeptimi itemq́ue
                  <lb/>
                rectangulum
                  <var>.q.n.</var>
                pro producto
                  <var>.d.c.</var>
                  <lb/>
                in
                  <var>.c.e.</var>
                & rectangulum
                  <var>.b.o.</var>
                ex
                  <var>.d.c.</var>
                in
                  <var>.
                    <lb/>
                  b.d.</var>
                & rectangulum
                  <var>.b.a.</var>
                ex
                  <var>.b.d.</var>
                in
                  <var>.d.
                    <lb/>
                  c.</var>
                et
                  <var>.p.f.</var>
                ex
                  <var>.d.b.</var>
                in
                  <var>.c.e.</var>
                ex .16. aut .20.
                  <lb/>
                prędictas. </s>
                <s xml:id="echoid-s748" xml:space="preserve">Quare ſex producta æquantur inter ſe,
                  <reg norm="replentque" type="simple">replentq́</reg>
                productum
                  <var>.u.s.</var>
                ex quo
                  <lb/>
                verum eſt propoſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div171" type="math:theorem" level="3" n="87">
              <head xml:id="echoid-head104" xml:space="preserve">THEOREMA
                <num value="87">LXXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s749" xml:space="preserve">QVA ratione cognoſci poſſit
                  <reg norm="verum" type="context">verũ</reg>
                eſſe proportionem ſummæ quatuor quan-
                  <lb/>
                titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-
                  <lb/>
                dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</s>
              </p>
              <p>
                <s xml:id="echoid-s750" xml:space="preserve">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-
                  <lb/>
                es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma </s>
              </p>
            </div>
          </div>
        </div>
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