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-div264" type="math:theorem" level="3" n="138">
              <p>
                <s xml:id="echoid-s1209" xml:space="preserve">
                  <pb o="93" rhead="THEOREM. ARIT." n="105" file="0105" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0105"/>
                tum eſt, ideo cognoſcemus
                  <var>.e.u.</var>
                ſed
                  <reg norm="cum" type="context">cũ</reg>
                  <var>.e.u.</var>
                minor ſit
                  <var>.a.u.</var>
                ex .18. & penultima primi,
                  <lb/>
                ſi
                  <reg norm="demptum" type="context">demptũ</reg>
                fuerit quadratum
                  <var>.e.u.</var>
                ex quadrato
                  <var>.a.u.</var>
                remanebit nobis
                  <reg norm="cognitum" type="context">cognitũ</reg>
                  <reg norm="quadra- tum" type="context">quadra-
                    <lb/>
                  tũ</reg>
                  <var>.a.e.</var>
                & ſic nota erit nobis perpendicularis
                  <var>.a.e.</var>
                ex penultima primi, quæ quidem
                  <var>.
                    <lb/>
                  a.e.</var>
                ſi multiplicata fuerit in dimidium
                  <var>.o.u.</var>
                dabit nobis
                  <reg norm="ſuperficiem" type="context">ſuperficiẽ</reg>
                trianguli
                  <var>.a.o.u.</var>
                ex
                  <lb/>
                41. dicti libri. </s>
                <s xml:id="echoid-s1210" xml:space="preserve">Et quia proportio trianguli
                  <var>.a.o.u.</var>
                ad triangulum
                  <var>.u.i.n.</var>
                (propter ſimi
                  <lb/>
                litudinem) eſt vt quadrati
                  <var>.o.u.</var>
                ad quadratum
                  <var>.n.i.</var>
                ex communi ſcientia cum vna-
                  <lb/>
                quæque iſtarum proportionum dupla ſit proportioni
                  <var>.o.u.</var>
                ad
                  <var>.n.i.</var>
                ex .17. et .18. ſexti,
                  <lb/>
                deinde cum nobis cognitæ ſint tres iſtarum quatuor quantitatum hoc eſt ſuperficies
                  <lb/>
                trianguli
                  <var>.a.o.u.</var>
                ſuperficies trianguli
                  <var>.u.n.i.</var>
                & quadrati
                  <var>.o.u.</var>
                </s>
                <s xml:id="echoid-s1211" xml:space="preserve">quare ex regula de tribus
                  <lb/>
                cognoſcemus etiam quadratum
                  <var>.n.i.</var>
                & ſic
                  <var>.n.i.</var>
                latus primi trianguli, vnde reliqua la
                  <lb/>
                tera illicò nobis innoteſcent exipſa regula de tribus, cum dixerimus, ſi
                  <var>.o.u.</var>
                dat nobis
                  <lb/>
                  <var>u.a</var>
                . </s>
                <s xml:id="echoid-s1212" xml:space="preserve">tunc
                  <var>.i.n.</var>
                dabit
                  <var>.u.n.</var>
                quòd etiam infero de
                  <var>.u.i</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1213" xml:space="preserve">Poſſemus etiam ita hoc perficere,
                  <lb/>
                ſcilicet inuenire
                  <var>.x.</var>
                quantitatem me-
                  <lb/>
                  <figure xlink:label="fig-0105-01" xlink:href="fig-0105-01a" number="144">
                    <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-01"/>
                  </figure>
                diam proportionalem inter duas ſu-
                  <lb/>
                perficies triangulorum, vnde ſuper-
                  <lb/>
                ficies trianguli
                  <var>.i.a.u.o.</var>
                ad
                  <var>.x.</var>
                ſe ha-
                  <lb/>
                beret ut
                  <var>.o.u.</var>
                ad
                  <var>.i.n.</var>
                & ita ex regula
                  <lb/>
                detribus cognoſcemus
                  <var>.i.n</var>
                . </s>
                <s xml:id="echoid-s1214" xml:space="preserve">Multo
                  <reg norm="tem" type="context">tẽ</reg>
                  <lb/>
                pore poſtquàm hoc theorema conſtruxi, ipſum conſcriptum inueni in decimo
                  <lb/>
                ſecundi libri Ioannis de monte Regio, ſatis tamen obſcurè expreſſum.</s>
              </p>
            </div>
            <div xml:id="echoid-div266" type="math:theorem" level="3" n="139">
              <head xml:id="echoid-head157" xml:space="preserve">THEOREMA
                <num value="139">CXXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s1215" xml:space="preserve">IN eodem primo libro vltimæ partis numerorum, Tartalea probat, via algebrę
                  <lb/>
                quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia,
                  <lb/>
                ſint tertio longiora per diame-
                  <lb/>
                  <figure xlink:label="fig-0105-02" xlink:href="fig-0105-02a" number="145">
                    <image file="0105-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0105-02"/>
                  </figure>
                trum circuli inſcriptibilis in ip-
                  <lb/>
                ſo triangulo. </s>
                <s xml:id="echoid-s1216" xml:space="preserve">ſed hoc breuius
                  <lb/>
                geometricè poteſt
                  <reg norm="demonſtrari" type="context">demõſtrari</reg>
                ,
                  <lb/>
                quemadmodum in ſubſcripta
                  <lb/>
                hic figura videre eſt, proptereà
                  <lb/>
                quòd cum anguli
                  <var>.A.o.u.</var>
                et
                  <var>.n.</var>
                  <lb/>
                omnes ſint recti et
                  <var>.A.u.</var>
                æqualis
                  <lb/>
                  <var>o.n.</var>
                et
                  <var>.A.n.</var>
                ęqualis
                  <var>.u.o.</var>
                ipſæ
                  <var>.A.
                    <lb/>
                  u.</var>
                et
                  <var>.A.n.</var>
                æquales erunt diame-
                  <lb/>
                tro ipſius circuli. </s>
                <s xml:id="echoid-s1217" xml:space="preserve">Sed eædem
                  <var>.
                    <lb/>
                  A.u.</var>
                et
                  <var>.A.n.</var>
                ſunt ſuperfluum, quo
                  <var>.A.B.</var>
                et
                  <var>.A.C.</var>
                ſunt maiores
                  <var>.B.C.</var>
                cum
                  <var>.B.u.</var>
                et
                  <var>.C.n.</var>
                  <lb/>
                ſint æquales
                  <var>.B.C.</var>
                ex penultima tertij Eucli.</s>
              </p>
              <head xml:id="echoid-head158" xml:space="preserve">THEO. SEQVENS THEO. CXXXIX.</head>
              <p>
                <s xml:id="echoid-s1218" xml:space="preserve">SImiliter in nono capite ſecundi libri nouæ ſcientiæ poterat ipſe Tartalea breuio
                  <lb/>
                ri methodo abſque vlla operatione ipſius Algebræ inuenire
                  <var>.A.H.</var>
                reſpectu
                  <var>.A.
                    <lb/>
                  E.</var>
                eſſe vt .4.
                  <reg norm="cum" type="context">cũ</reg>
                vno ſeptimo ad
                  <reg norm="vnum" type="context">vnũ</reg>
                . </s>
                <s xml:id="echoid-s1219" xml:space="preserve">
                  <reg norm="Nam" type="context">Nã</reg>
                ipſe ſupponit
                  <var>.A.E.</var>
                  <reg norm="decimam" type="context">decimã</reg>
                  <reg norm="partem" type="context">partẽ</reg>
                eſſe ipſius </s>
              </p>
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