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. ARIT.
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            <div xml:id="echoid-div146" type="math:theorem" level="3" n="74">
              <p>
                <s xml:id="echoid-s642" xml:space="preserve">
                  <pb o="49" rhead="THEOREM. ARIT." n="61" file="0061" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0061"/>
                Secundus tertiusq́ue terminus reperiuntur, eſt
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/>
                enim ſecundus
                  <var>.e.i.</var>
                tertius
                  <var>.i.o.</var>
                et
                  <var>.e.a.</var>
                quando-
                  <lb/>
                quidem ex præſuppoſito
                  <var>.e.i.</var>
                æqualis eſt
                  <var>.s.q.</var>
                et
                  <lb/>
                  <var>i.o.</var>
                æqualis
                  <var>.r.c.</var>
                et
                  <var>.a.e.</var>
                cum ſit æqualis
                  <var>.g.t.</var>
                cui
                  <lb/>
                pariter æqualis eſt
                  <var>.r.u.</var>
                ex quo
                  <var>.a.e.</var>
                æqualis
                  <lb/>
                eſt
                  <var>.u.r</var>
                . </s>
                <s xml:id="echoid-s643" xml:space="preserve">Itaque illud ſequitur
                  <var>.a.o.</var>
                ipſi
                  <var>.q.p.</var>
                  <lb/>
                æqualem eſſe.</s>
              </p>
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                <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a">
                  <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-01"/>
                </figure>
              </div>
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            <div xml:id="echoid-div148" type="math:theorem" level="3" n="75">
              <head xml:id="echoid-head91" xml:space="preserve">THEOREMA
                <num value="75">LXXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s644" xml:space="preserve">CVR ſumma duorum terminorum extremorum imparium arithmeticæ pro-
                  <lb/>
                portionalitatis ſemper duplo medij termini æqualis eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s645" xml:space="preserve">Exempli gratia, ſunt hitres termini proportionalitatis arithmeticæ .20. 15. 10
                  <lb/>
                ſumma duorum extremorum erit .30. quæ duplum eſt medij termini .15.</s>
              </p>
              <p>
                <s xml:id="echoid-s646" xml:space="preserve">Quod vt ſpeculemur, tres termini, tribus lineis
                  <var>.b.d</var>
                :
                  <var>n.u.</var>
                et
                  <var>.q.p.</var>
                  <reg norm="ſignificentur" type="context">ſignificẽtur</reg>
                . </s>
                <s xml:id="echoid-s647" xml:space="preserve">Di-
                  <lb/>
                co nunc quòd ſumma
                  <var>.b.d.</var>
                cum
                  <var>.q.p.</var>
                nempe
                  <var>.
                    <lb/>
                  h.d.</var>
                ſemper duplo
                  <var>.n.u.</var>
                ſcilicet
                  <var>.g.u.</var>
                æqualis
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0061-02a" xlink:href="fig-0061-02"/>
                erit. </s>
                <s xml:id="echoid-s648" xml:space="preserve">Tum differentia
                  <var>.b.d.</var>
                ad
                  <var>.n.u.</var>
                ſit
                  <var>.c.d.</var>
                quæ
                  <lb/>
                æqualis erit
                  <var>.e.u.</var>
                differentiæ inter
                  <var>n.u.</var>
                et
                  <var>.q.p.</var>
                  <lb/>
                patet enim in linea
                  <var>.h.d</var>
                :
                  <var>b.c.</var>
                æqualem eſſe
                  <var>.n.
                    <lb/>
                  u.</var>
                ſed
                  <var>.n.u.</var>
                ex
                  <var>.n.e.</var>
                componitur æquali
                  <var>.q.p.</var>
                et
                  <lb/>
                ex
                  <var>.e.u.</var>
                æquali
                  <var>.c.d.</var>
                cum
                  <reg norm="itaque" type="simple">itaq;</reg>
                in
                  <var>.h.d.</var>
                partem
                  <var>.
                    <lb/>
                  h.b.</var>
                reperiamus æqualem
                  <var>.n.e.</var>
                gratia
                  <var>.q.p.</var>
                &
                  <lb/>
                partem
                  <var>.c.d.</var>
                æquale
                  <var>m.e.u.</var>
                manifeſtum erit
                  <lb/>
                  <var>h.d.</var>
                æqualem eſſe
                  <var>.g.u</var>
                .</s>
              </p>
              <div xml:id="echoid-div148" type="float" level="4" n="1">
                <figure xlink:label="fig-0061-02" xlink:href="fig-0061-02a">
                  <image file="0061-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-02"/>
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              </div>
              <head xml:id="echoid-head92" style="it" xml:space="preserve">BINA PROBLEMAT A EX DVOBVS PRAEDICTIS
                <lb/>
              THEOREMATIBVS DEPENDENTIA.</head>
              <p>
                <s xml:id="echoid-s649" xml:space="preserve">EX duobus prædictis theorematibus duo problemata oriuntur,
                  <reg norm="quorum" type="context">quorũ</reg>
                primum
                  <lb/>
                eſt. </s>
                <s xml:id="echoid-s650" xml:space="preserve">Datis tribus quantitatibus cognitis, ſi quis quartam inuenire voluerit,
                  <lb/>
                quæ eiuſmodi ſit reſpectu tertiæ, qualis eſt ſecunda reſpectu primæ, ſecunda cum
                  <lb/>
                tertia in ſummam colligenda erit, ex qua detracta prima, ſupererit quarta.</s>
              </p>
              <p>
                <s xml:id="echoid-s651" xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. ſi quartam inuenire vo
                  <lb/>
                luerimus eiuſmodi proportionem cum tertia arithmeticè ſeruantem, quam ſecunda
                  <lb/>
                cum prima, ſecundam cum tertia in ſummam colligemus,
                  <reg norm="dabiturque" type="simple">dabiturq́;</reg>
                ſumma .26. ex
                  <lb/>
                qua detracta prima quantitate, quarta relinquetur nempe .6. quod ex .74. theore-
                  <lb/>
                mate dependet.</s>
              </p>
              <p>
                <s xml:id="echoid-s652" xml:space="preserve">Idipſum tamen proueniret ſi quis ex tertio termino differentiam primi atque ſe-
                  <lb/>
                cundi detraheret; </s>
                <s xml:id="echoid-s653" xml:space="preserve">hæc tamen via non tam vniuerſalis eſtqu àm illa. </s>
                <s xml:id="echoid-s654" xml:space="preserve">N ſi quartus ter
                  <lb/>
                minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-
                  <lb/>
                mino in ſummam colligere oporteret.</s>
              </p>
              <p>
                <s xml:id="echoid-s655" xml:space="preserve">Alterum problema eſt, quòd inuentis duobus terminis, ſi tertius requiratur, ſe-
                  <lb/>
                cundus duplicandus erit, ex qua ſumma detracto primo, ſtatim tertius proferetur,
                  <lb/>
                quod problema ex præcedenti theoremate dependet.</s>
              </p>
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