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-div63" type="math:theorem" level="3" n="28">
              <p>
                <s xml:id="echoid-s275" xml:space="preserve">
                  <pb o="19" rhead="THEOREM. ARIT." n="31" file="0031" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0031"/>
                bit quadratum
                  <var>.e.d.</var>
                cognitum, cuius radix æqualis erit
                  <var>.c.t.</var>
                qua coniuncta dimi-
                  <lb/>
                dio
                  <var>.c.a.</var>
                ex quinta ſecundi Eucli. dabit quod propoſitum erat.</s>
              </p>
            </div>
            <div xml:id="echoid-div65" type="math:theorem" level="3" n="29">
              <head xml:id="echoid-head45" xml:space="preserve">THEOREMA
                <num value="29">XXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s276" xml:space="preserve">
                  <emph style="sc">QVid</emph>
                cauſæ eſt, cur ſubtracto duplo producti duorum numerorum ad inui-
                  <lb/>
                cem
                  <reg norm="multiplicatorum" type="context">multiplicatorũ</reg>
                ex ſumma ſuorum quadratorum, ſemper quod ſuper
                  <lb/>
                eſt duorum numerorum quadratum differentiæ ſit?</s>
              </p>
              <p>
                <s xml:id="echoid-s277" xml:space="preserve">Exempli gratia ſi proponerentur duo numeri .16. et .4. duplum producti eorum
                  <lb/>
                eſſet .128. quò detracto ex ſumma ſuorum quadratorum, nempè ex .272. rema-
                  <lb/>
                neret .144. cuius quadrati radix eſſet .12. tanquam differentia inter .4. et .16.</s>
              </p>
              <p>
                <s xml:id="echoid-s278" xml:space="preserve">Id vtſciamus, duo numeri propoſiti, duabus lineis ſignificentur, maiore
                  <var>.q.g.</var>
                  <lb/>
                et minore
                  <var>.g.p.</var>
                directè coniunctis, ſuper quas, totale quadratum extruatur
                  <var>.a.p.</var>
                  <lb/>
                in quo cogitetur diameter
                  <var>.a.p.</var>
                et à puncto
                  <var>.g.</var>
                ducatur parallela
                  <var>.g.n.c.</var>
                et à pun-
                  <lb/>
                cto
                  <var>.n.</var>
                parallela
                  <var>.n.s.r.</var>
                ex quo duo producta
                  <reg norm="dabuntur" type="context">dabũtur</reg>
                  <var>.q.n.</var>
                et
                  <var>.n.u.</var>
                ſingula æqualia pro-
                  <lb/>
                ducto
                  <var>.q.g.</var>
                in
                  <var>g.p.</var>
                et
                  <var>.a.n.</var>
                et
                  <var>.n.p.</var>
                duo quadrata dictorum numerorum propoſi-
                  <lb/>
                torum, quod ſatis
                  <reg norm="ſuperque" type="simple">ſuperq́</reg>
                , probatur quarta ſecundi Eucli. </s>
                <s xml:id="echoid-s279" xml:space="preserve">Cogitemus deinde
                  <var>.n.
                    <lb/>
                  o.</var>
                æqualem
                  <var>.n.p.</var>
                et à puncto
                  <var>.o.</var>
                ducatur
                  <var>.o.m.t.</var>
                parallela
                  <var>.r.s.</var>
                et
                  <var>.o.e.</var>
                ad
                  <var>.n.
                    <lb/>
                  c</var>
                . </s>
                <s xml:id="echoid-s280" xml:space="preserve">quare ex allatis ab Eucli. octaua ſecundi, dabi-
                  <lb/>
                tur quantitas
                  <var>.m.n.</var>
                æqualis
                  <var>.q.n.</var>
                producto
                  <var>.q.g.</var>
                in
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/>
                  <var>g.p.</var>
                et quantitas
                  <var>.o.c.</var>
                minor ipſo producto, ex
                  <lb/>
                quantitate quadrati
                  <var>.n.p.</var>
                ex quo quantitas
                  <var>.m.n.e.</var>
                  <lb/>
                vna cum quadrato
                  <var>.n.p.</var>
                æqualis erit duplo produ-
                  <lb/>
                cti
                  <var>.q.g.</var>
                in
                  <var>.g.p.</var>
                ſed hæ duæ quantitates, ſunt par-
                  <lb/>
                tes duorum quadratorum dictorum, & quæ ſuper
                  <lb/>
                eſt
                  <var>.m.e.</var>
                quadratum differentiæ vnius numeri pro-
                  <lb/>
                poſiti ab altero, prout in ſubſcripta figura licebit cui
                  <lb/>
                libet conſiderare. </s>
                <s xml:id="echoid-s281" xml:space="preserve">Itaque veritas hæc manifeſta
                  <lb/>
                erit.</s>
              </p>
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                <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a">
                  <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div67" type="math:theorem" level="3" n="30">
              <head xml:id="echoid-head46" xml:space="preserve">THEOREMA
                <num value="30">XXX</num>
              .</head>
              <p>
                <s xml:id="echoid-s282" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                ij qui ex duobus numeris propoſitis maiorem per minorem diuidunt, ſi
                  <lb/>
                proueniens per maiorem numerum multiplicauerint, productum æquale
                  <lb/>
                erit prouenienti ex diuiſione quadrati maioris numeri per minorem?</s>
              </p>
              <p>
                <s xml:id="echoid-s283" xml:space="preserve">Exempli gratia ſi proponantur duo numeri .20. et .4.
                  <reg norm="ipſeque" type="simple">ipſeq́</reg>
                .20. per .4. diui-
                  <lb/>
                datur, dabit quinque, tum .400. quadrato .20. diuiſo per prioré .4. dabit .100.
                  <lb/>
                quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s284" xml:space="preserve">Cuius ſpeculationis cauſa, ſint duo numeri, qui lineis
                  <var>.x.u.</var>
                et
                  <var>.x.s.</var>
                maiore
                  <reg norm="atque" type="simple">atq;</reg>
                mi-
                  <lb/>
                nore ſignificétur, tum
                  <var>.u.x.</var>
                numerus per
                  <var>.s.x.</var>
                di-
                  <lb/>
                uidatur, ſitq́ue proueniens
                  <var>.x.n.</var>
                poſtmodum qua-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/>
                dratum
                  <var>.u.x.</var>
                ſit
                  <var>.x.o.</var>
                et productum ex
                  <var>.n.x.</var>
                in
                  <var>.u.
                    <lb/>
                  x.</var>
                ſit
                  <var>.x.e.</var>
                quod æquale eſſe dico prouenienti ex
                  <lb/>
                diuiſione quadrati
                  <var>.o.x.</var>
                per
                  <var>.s.x.</var>
                quod ſit
                  <var>.m</var>
                . </s>
                <s xml:id="echoid-s285" xml:space="preserve">Patet
                  <lb/>
                enim ex definitione diuiſionis, talem futuram pro-
                  <lb/>
                portionem
                  <var>.u.x.</var>
                ad
                  <var>.n.x.</var>
                qualis eſt
                  <var>.s.x.</var>
                ad vnitatem,
                  <lb/>
                & quadratum
                  <var>.o.x.</var>
                ad rectangulum
                  <var>.e.x.</var>
                ita ſe ha- </s>
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