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-div159" type="math:theorem" level="3" n="81">
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            <div xml:id="echoid-div161" type="math:theorem" level="3" n="82">
              <head xml:id="echoid-head99" xml:space="preserve">THEOREMA
                <num value="82">LXXXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s717" xml:space="preserve">CVR quantitate aliqua in quatuor partes
                  <reg norm="continuas" type="context">cõtinuas</reg>
                proportionales ſecta per-
                  <lb/>
                q́ue ſingulas diuiſa, ſumma quatuor prouenientium æqualis ſit producto ſe-
                  <lb/>
                cundi in tertium.</s>
              </p>
              <p>
                <s xml:id="echoid-s718" xml:space="preserve">Exempli gratia, ſi triginta in quatuor partes proportionales ſecetur, hoc eſt.
                  <lb/>
                16. 8. 4. 2.
                  <reg norm="perque" type="simple">perq́;</reg>
                harum ſingulas idem numerus .30. diuidatur, primum proueniens
                  <lb/>
                erit .1. cum ſeptem octauis partibus. </s>
                <s xml:id="echoid-s719" xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7.
                  <lb/>
                cum dimidio, quartum .15. integri, quorum ſumma erit .28. cum octaua parte, tan
                  <lb/>
                  <reg norm="tumque" type="simple">tumq́;</reg>
                erit productum ſecundi prouenientis in tertium.</s>
              </p>
              <p>
                <s xml:id="echoid-s720" xml:space="preserve">Quod vt ſciamus, quantitas
                  <var>.n.c.</var>
                in partes continuas proportionales quatuor ſe-
                  <lb/>
                cetur
                  <var>.n.a</var>
                :
                  <var>a.t</var>
                :
                  <var>t.e.</var>
                et
                  <var>.e.c.</var>
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                per ſingulas partes illa ipſa diuiſa, prouenientia
                  <lb/>
                ſint
                  <var>.i.d</var>
                :
                  <var>d.x</var>
                :
                  <var>x.u</var>
                :
                  <var>u.o.</var>
                  <reg norm="quorum" type="context">quorũ</reg>
                ſumma ſit
                  <var>.i.o.</var>
                hanc
                  <reg norm="ſummam" type="context">ſummã</reg>
                dicimus æqualem eſſe nume-
                  <lb/>
                ro producti
                  <var>.d.x.</var>
                in
                  <var>.x.u</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s721" xml:space="preserve">Quod hac ratione probo, cogito productam eſſe lineam
                  <var>.i.o.</var>
                  <reg norm="quousque" type="simple">quousq́;</reg>
                  <var>.o.p.</var>
                æqua
                  <lb/>
                lis ſit
                  <var>.o.u.</var>
                  <reg norm="erectamque" type="simple">erectamq́;</reg>
                  <var>.m.o.</var>
                æqualem
                  <var>.i.d.</var>
                perpendiculariter
                  <var>.o.p.</var>
                & productam donec
                  <var>.
                    <lb/>
                  o.q.</var>
                vnitati ſit æqualis. </s>
                <s xml:id="echoid-s722" xml:space="preserve">Iam terminatis rectangulis
                  <var>.m.p.</var>
                et
                  <var>.i.q.</var>
                patebit ex .15. ſexti
                  <lb/>
                aut .20. ſeptimi, productum
                  <var>.m.p.</var>
                producto
                  <var>.d.x.</var>
                in
                  <var>.x.u.</var>
                æquale eſſe. </s>
                <s xml:id="echoid-s723" xml:space="preserve">Ita quòd ſi pro-
                  <lb/>
                bauero productum
                  <var>.i.q.</var>
                producto
                  <var>.m.p.</var>
                æquale eſſe, facile patebit propoſitum. </s>
                <s xml:id="echoid-s724" xml:space="preserve">Cuius
                  <lb/>
                gratia, ſequuti præcedentis theorematis ordinem, primum ex
                  <reg norm="definitionem" type="context">definitionẽ</reg>
                diuiſionis,
                  <lb/>
                eadem proportio erit
                  <var>.n.c.</var>
                ad
                  <var>.i.d.</var>
                quæ
                  <var>.n.a.</var>
                ad
                  <var>.o.q.</var>
                ex quo permutando
                  <var>.n.c.</var>
                ad
                  <var>.n.a.</var>
                ſic
                  <lb/>
                ſe habebit vt
                  <var>.i.d.</var>
                hoc eſt
                  <var>.m.o.</var>
                ad
                  <var>.o.q.</var>
                & ſi progrediamur eodem ordine, quo præ-
                  <lb/>
                cedenti theoremate, ſumpto principio ab
                  <var>.i.d.</var>
                et
                  <var>.e.c.</var>
                verſus
                  <var>.d.x.</var>
                et
                  <var>.e.t.</var>
                gradatimq́ue
                  <lb/>
                permutando ac coniungendo, inue-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/>
                niemus eandem proportionem eſſe
                  <lb/>
                  <var>c.n.</var>
                ad
                  <var>.n.a.</var>
                quæ
                  <var>.i.o.</var>
                ad
                  <var>.o.u.</var>
                nempe
                  <var>.
                    <lb/>
                  o.p.</var>
                ex quo ex .11 quinti, ita ſe habe
                  <lb/>
                bit
                  <var>.i.o.</var>
                ad
                  <var>.o.p.</var>
                vt
                  <var>.m.o.</var>
                ad
                  <var>.o.q.</var>
                </s>
                <s xml:id="echoid-s725" xml:space="preserve">quare
                  <lb/>
                ex .15. ſextiaut .20. ſeptimi
                  <reg norm="produ- ctum" type="context">produ-
                    <lb/>
                  ctũ</reg>
                  <var>.i.q.</var>
                erit producto
                  <unsure/>
                  <var>.m.p.</var>
                æquale,
                  <lb/>
                ex quo etiam æquale erit producto
                  <var>.
                    <lb/>
                  d.x.</var>
                in
                  <var>.x.u</var>
                . </s>
                <s xml:id="echoid-s726" xml:space="preserve">Idem ordo in qualibet
                  <lb/>
                quantitate in quantaſuis partes diuiſa ſeruari poterit, cum huiuſmodi
                  <reg norm="ſcientia" type="context">ſciẽtia</reg>
                in vni
                  <lb/>
                uerſum pateat.</s>
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                <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a">
                  <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0066-01"/>
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              </div>
            </div>
            <div xml:id="echoid-div163" type="math:theorem" level="3" n="83">
              <head xml:id="echoid-head100" xml:space="preserve">THEOREMA
                <num value="83">LXXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s727" xml:space="preserve">CVR termini medij cubus, trium continuè proportionalium, ſemper producto
                  <lb/>
                rectanguli compræhenſi à maximo & medio in minimo termino æqualis ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s728" xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. ſi
                  <lb/>
                ſumpſerimus productum maximi in medium nempe .54. quod per
                  <reg norm="minimum" type="context">minimũ</reg>
                .4. multi-
                  <lb/>
                plicemus, dabitur numerus .216. cubo medij .6. æqualis.</s>
              </p>
              <p>
                <s xml:id="echoid-s729" xml:space="preserve">In cuius gratiam tres numeri continui proportionales tribus lineis
                  <var>.a.e.i.</var>
                  <reg norm="ſignifi- centur" type="context">ſignifi-
                    <lb/>
                  cẽtur</reg>
                , cubus autem
                  <var>.e.</var>
                ſignificetur figura
                  <var>.d.n.</var>
                  <reg norm="productumque" type="simple">productumq́</reg>
                  <var>.a.</var>
                in
                  <var>.e.</var>
                ſit
                  <var>.b.n.</var>
                ipſius
                  <reg norm="au- temmet" type="context">au-
                    <lb/>
                  tẽmet</reg>
                in
                  <var>.i.</var>
                ſit
                  <var>.p.o.</var>
                ita quod
                  <var>.q.p.</var>
                aut
                  <var>.b.o.</var>
                cum ſint
                  <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                ſpeciei, æqualis erit .a: et
                  <var>.o.n.</var>
                </s>
              </p>
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