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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THEOR. ARITH.
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              <p>
                <s xml:id="echoid-s197" xml:space="preserve">
                  <pb o="13" rhead="THEOR. ARITH." n="25" file="0025" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0025"/>
                tum ipſius
                  <var>.d.q.</var>
                talem eſſe partem quadrati ipſius
                  <var>.b.q.</var>
                qualis quadratum ipſius
                  <var>.g.i.</var>
                  <lb/>
                eſt quadrati ipſius
                  <var>.f.g</var>
                . </s>
                <s xml:id="echoid-s198" xml:space="preserve">Scimus pręterea ex .19. ſexti, aut vndecima octaui, propor-
                  <lb/>
                tioné quadrati ipſius
                  <var>.b.q.</var>
                ad
                  <reg norm="quadratum" type="context">quadratũ</reg>
                ipſius
                  <var>.d.q.</var>
                duplam eſſe proportioni
                  <var>.b.q.</var>
                ad
                  <var>.
                    <lb/>
                  d.q.</var>
                ſuarum radicum (cuborum enim tripla eſſet & cenſuum cenſuum, quadrupla,
                  <lb/>
                  <reg norm="atque" type="simple">atq;</reg>
                ita deinceps ex præcedenti theoremate) Id ipſum dico de dignitatibus ipſius
                  <var>.
                    <lb/>
                  f.g.</var>
                et
                  <var>.i.g.</var>
                reſpectu radicum
                  <var>.f.g.</var>
                et
                  <var>.i.g</var>
                . </s>
                <s xml:id="echoid-s199" xml:space="preserve">Vnde
                  <lb/>
                cum proportio dignitatis ipſius
                  <var>.b.q.</var>
                ad il-
                  <lb/>
                lam
                  <var>.d.q.</var>
                ęqualis ſit proportioni dignitatis
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/>
                ipſius
                  <var>.f.g.</var>
                ad illam
                  <var>.g.i.</var>
                ex communi ſcien-
                  <lb/>
                tia apertè cognoſcemus ſimplices propor-
                  <lb/>
                tiones eſſe interſe æquales, nempe eam quę
                  <lb/>
                eſt
                  <var>.b.q.</var>
                ad
                  <var>.d.q.</var>
                æqualem eſſe ei, quæ eſt
                  <var>.f.
                    <lb/>
                  g.</var>
                ad
                  <var>.i.g.</var>
                  <reg norm="itaque" type="simple">itaq;</reg>
                ſequitur ex definitione diuiſionis
                  <var>.d.q.</var>
                eſſe proueniens ex diuiſione
                  <var>.
                    <lb/>
                  b.q.</var>
                per
                  <var>.f.g</var>
                .</s>
              </p>
              <div xml:id="echoid-div42" type="float" level="4" n="1">
                <figure xlink:label="fig-0025-01" xlink:href="fig-0025-01a">
                  <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-01"/>
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              </div>
            </div>
            <div xml:id="echoid-div44" type="math:theorem" level="3" n="19">
              <head xml:id="echoid-head35" xml:space="preserve">THEOREMA
                <num value="19">XVIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s200" xml:space="preserve">CVR productum ex duabus radicibus quadratis, eſt quadrata radix, producti
                  <lb/>
                ſuorum quadratorum ſimul?</s>
              </p>
              <p>
                <s xml:id="echoid-s201" xml:space="preserve">In cuius rei gratiam, ſint duo quadrata
                  <var>.d.a.</var>
                et
                  <var>n.o.</var>
                coniuncta ſimul, prout in ſub-
                  <lb/>
                ſcripta figura apparet, ita tamen vtangulus
                  <var>.a.n.u.</var>
                ſitre
                  <lb/>
                ctus, </s>
                <s xml:id="echoid-s202" xml:space="preserve">quare ex quartadecima primi, duo latera
                  <var>.n.c.</var>
                et
                  <var>.
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0025-02a" xlink:href="fig-0025-02"/>
                  n.a.</var>
                directe
                  <reg norm="coniungentur" type="context">coniũgentur</reg>
                adinuicem, prout etiam reli-
                  <lb/>
                qua duo latera
                  <var>.n.u.</var>
                et
                  <var>.n.d</var>
                . </s>
                <s xml:id="echoid-s203" xml:space="preserve">Cogitato deinde
                  <var>.a.u.</var>
                pro
                  <lb/>
                ducto ipſius
                  <var>.a.n.</var>
                in
                  <var>.n.u.</var>
                duarum videlicet radicum
                  <lb/>
                quadratarum ſimul, dabitur ex prima ſexti, aut de-
                  <lb/>
                cimaottaua ſeptimi, productum
                  <var>.a.u.</var>
                medium propor
                  <lb/>
                tionale inter quadratum
                  <var>.a.d.</var>
                et
                  <var>.u.c.</var>
                quod ſi cogi-
                  <lb/>
                temus has tres ſuperficies, tres numeros eſſe, pate-
                  <lb/>
                bit ex vigeſimaprima ſeptimi productum
                  <var>.a.u.</var>
                in ſe-
                  <lb/>
                ipſum, quadratum ſcilicet
                  <var>.a.u.</var>
                æquale eſſe producto
                  <var>.
                    <lb/>
                  a.d.</var>
                in
                  <var>.u.c.</var>
                ex quo propoſiti euidentia conſequetur.</s>
              </p>
              <div xml:id="echoid-div44" type="float" level="4" n="1">
                <figure xlink:label="fig-0025-02" xlink:href="fig-0025-02a">
                  <image file="0025-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div46" type="math:theorem" level="3" n="20">
              <head xml:id="echoid-head36" xml:space="preserve">THEOREMA
                <num value="20">XX</num>
              .</head>
              <p>
                <s xml:id="echoid-s204" xml:space="preserve">QVA ratione id ipſum in cubis cognoſci poterit.
                  <lb/>
                </s>
                <s xml:id="echoid-s205" xml:space="preserve">Sit cubus
                  <var>.l.b.</var>
                & cubus
                  <var>.o.p.</var>
                quorum productum ſit
                  <var>.u.g.</var>
                quod aſſero eſle
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0025-03a" xlink:href="fig-0025-03"/>
                cubum, quamuis Eucli. idem probet
                  <lb/>
                in
                  <ref id="ref-0008">.4. noni.</ref>
                cuius radicem demonſtra-
                  <lb/>
                bo eſſe numeri æqualis numero
                  <var>.m.q.</var>
                  <lb/>
                qui
                  <var>.m.q.</var>
                productum eſt ipſius
                  <var>.m.e.</var>
                in
                  <var>.e.
                    <lb/>
                  q.</var>
                radicum propoſitorum cuborum. </s>
                <s xml:id="echoid-s206" xml:space="preserve">Pa-
                  <lb/>
                tet enim ex præcedenti theoremate
                  <var>.m.
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0025-04a" xlink:href="fig-0025-04"/>
                  </var>
                </s>
              </p>
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