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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THEOREM. ARITH.
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            <div xml:id="echoid-div171" type="math:theorem" level="3" n="87">
              <p>
                <s xml:id="echoid-s750" xml:space="preserve">
                  <pb o="57" rhead="THEOREM. ARITH." n="69" file="0069" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0069"/>
                primæ cum tertia .20. ex quo ſicſe habet .20. ad .8. nempe ad ſecundam, vt .30.
                  <lb/>
                ad .12.</s>
              </p>
              <p>
                <s xml:id="echoid-s751" xml:space="preserve">Quod vt ſciamus, quatuor prædictæ quantitates ſignificentur linea
                  <var>.a.e.i.o.</var>
                pro-
                  <lb/>
                babo ita ſe habere
                  <var>.a.e.i.o.</var>
                ad
                  <var>.e.i.</var>
                vt
                  <var>.a.i.</var>
                ad
                  <var>.e</var>
                . </s>
                <s xml:id="echoid-s752" xml:space="preserve">Nam cum ſic ſe habeat
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                ut
                  <var>.e.</var>
                  <lb/>
                ad
                  <var>.i.</var>
                & vt
                  <var>.i.</var>
                ad .o: ex æqualitate proportionum vel permutando ita ſe habebit
                  <var>.a.</var>
                ad
                  <var>.i.</var>
                  <lb/>
                vt
                  <var>.e.</var>
                ad
                  <var>.o.</var>
                & è conuerſo ita
                  <var>.o.</var>
                ad
                  <var>.e.</var>
                vt
                  <var>.i.</var>
                ad
                  <var>.a.</var>
                &
                  <reg norm="componendo" type="context">cõponendo</reg>
                ita
                  <var>.o.e.</var>
                ad e. vt
                  <var>.i.a.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                  <reg norm="permutandoque" type="simple">permutandoq́</reg>
                  <var>.o.e.</var>
                ad
                  <var>.i.a.</var>
                vt
                  <var>.e.</var>
                ad
                  <var>.a.</var>
                nempe
                  <var>.i.</var>
                ad
                  <var>.e.</var>
                & componendo ita
                  <var>.o.i.e.a.</var>
                ad
                  <var>.
                    <lb/>
                  i.a.</var>
                vt
                  <var>.i.e.</var>
                ad
                  <var>.e.</var>
                & permutando ita
                  <var>.o.i.e.a.</var>
                ad
                  <var>.i.e.</var>
                vt
                  <var>.i.a.</var>
                ad
                  <var>.e.</var>
                quod erat propoſitum.
                  <lb/>
                </s>
                <s xml:id="echoid-s753" xml:space="preserve">Ex quo patet error antiquorum quiidipſum, accidere arbitrati ſunt in quantitatibus
                  <lb/>
                diſcretæ proportionalitatis, quod tamen falſum eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s754" xml:space="preserve">Exempli gratia, ſi proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eſt quæ .4.
                  <lb/>
                ad .2. </s>
                <s xml:id="echoid-s755" xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non ſit eadem quæ .12. ad .6. harum
                  <lb/>
                autem ſumma erit .24. & ſumma ſecundæ cum tertia .10. ſed primæ cum tertia erit
                  <lb/>
                16. ex quo .16. ad .6. non ſic ſe habebit vt .24. ad .10.
                  <lb/>
                At in ſpeculatione quatuor quantitatum
                  <var>.a.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/>
                  <var>e.i.o.</var>
                ſi proportio
                  <var>.e.</var>
                ad
                  <var>.i.</var>
                non eſſet eadem
                  <lb/>
                quæ
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                minimè licuiſſet dicere ita ſe
                  <lb/>
                habere
                  <var>.i.</var>
                ad
                  <var>.e.</var>
                vt
                  <var>.e.</var>
                ad
                  <var>.a</var>
                .</s>
              </p>
              <div xml:id="echoid-div171" type="float" level="4" n="1">
                <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a">
                  <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div173" type="math:theorem" level="3" n="88">
              <head xml:id="echoid-head105" xml:space="preserve">THEOREMA
                <num value="88">LXXXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s756" xml:space="preserve">CVR extribus quantitatibus quibuſlibet, productum duarum in tertiam, vna
                  <lb/>
                ſemper
                  <reg norm="eademque" type="simple">eademq́;</reg>
                ſit quantitas.</s>
              </p>
              <p>
                <s xml:id="echoid-s757" xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. ſi multiplicauerimus .15. per .8. tum produ
                  <lb/>
                ctum per .2. tantum erit quantum ſi quis multiplicaret .8. per .2. & hoc per .15. et .15.
                  <lb/>
                per .2.
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                per .8.</s>
              </p>
              <p>
                <s xml:id="echoid-s758" xml:space="preserve">Quod ut pateat, tres quantitates tri-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0069-02a" xlink:href="fig-0069-02"/>
                bus lineis ſignificentur
                  <var>.m.f</var>
                : a. et
                  <var>.o</var>
                . </s>
                <s xml:id="echoid-s759" xml:space="preserve">Dico
                  <lb/>
                productum
                  <var>.m.f.</var>
                in
                  <var>.a.</var>
                multiplicatum.
                  <lb/>
                </s>
                <s xml:id="echoid-s760" xml:space="preserve">per
                  <var>.o.</var>
                æquale eſſe producto
                  <var>.a.</var>
                in
                  <var>.o.</var>
                mul-
                  <lb/>
                tiplicato per
                  <var>.m.f.</var>
                aut producto
                  <var>.m.f.</var>
                in
                  <var>.
                    <lb/>
                  o.</var>
                multiplicato per
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s761" xml:space="preserve">Sit enim corpus
                  <var>.d.
                    <lb/>
                  u.</var>
                  <reg norm="rectangulum" type="context">rectãgulum</reg>
                , cuius latus
                  <var>.n.u.</var>
                ſit æquale
                  <lb/>
                  <var>m.f.</var>
                et
                  <var>.u.t</var>
                : a: et
                  <var>.u.c</var>
                : o. patebit manifeſtè
                  <lb/>
                  <var>n.t.</var>
                eſſe productum
                  <var>.m.f.</var>
                in
                  <var>.a.</var>
                quod
                  <var>.n.t.</var>
                  <lb/>
                multiplicatum in
                  <var>.u.c.</var>
                æquali
                  <var>.o.</var>
                producit
                  <lb/>
                corpus
                  <var>.d.u.</var>
                ſed idipſum corpus
                  <var>.d.u.</var>
                ex
                  <lb/>
                multiplicatione producti
                  <var>.c.t.</var>
                in latus
                  <var>.n.
                    <lb/>
                  u.</var>
                æquale
                  <var>.m.f.</var>
                oritur, & idipſum
                  <var>.d.u.</var>
                ex
                  <lb/>
                multiplicatione
                  <var>.n.c.</var>
                in latus
                  <var>.u.t.</var>
                æquale
                  <var>.a.</var>
                profertur.</s>
              </p>
              <div xml:id="echoid-div173" type="float" level="4" n="1">
                <figure xlink:label="fig-0069-02" xlink:href="fig-0069-02a">
                  <image file="0069-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div175" type="math:theorem" level="3" n="89">
              <head xml:id="echoid-head106" xml:space="preserve">THEOREMA
                <num value="89">LXXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s762" xml:space="preserve">CVR quarumcunque quatuor quantitatum, ſi prima in ſecundam multiplice-
                  <lb/>
                tur & hoc productum in tertiam,
                  <reg norm="rurſusque" type="simple">rurſusq́</reg>
                hoc alterum in quartam, vltimum
                  <lb/>
                productum æquale ſit producto producti ſecundæ in tertiam, in productum primæ
                  <lb/>
                in quartam.</s>
              </p>
            </div>
          </div>
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