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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            <div xml:id="echoid-div94" type="math:theorem" level="3" n="44">
              <p>
                <s xml:id="echoid-s389" xml:space="preserve">
                  <pb o="29" rhead="THEOR. ARITH." n="41" file="0041" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0041"/>
                to maiore
                  <var>.c.t.</var>
                extractum quare reſiduum qua-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/>
                drati
                  <var>.c.p.</var>
                cognitum erit, quam quantitatem co-
                  <lb/>
                gnitam, cum ſit ſecundo loco data, cogitemus
                  <lb/>
                detrahi è toto quadrato cognito
                  <var>.q.e.</var>
                ex quo
                  <lb/>
                ſumma duorum ſupplementorum
                  <var>.q.o.</var>
                et
                  <var>.o.e.</var>
                  <lb/>
                cognoſcetur, vnà cum quadratis
                  <var>.u.n.</var>
                et
                  <var>.p.a.</var>
                du
                  <lb/>
                plo ſcilicet
                  <var>.q.a.</var>
                quo diuiſo per duplum
                  <var>.q.h.</var>
                aut
                  <lb/>
                ſimplex
                  <var>.q.a.</var>
                per
                  <var>.q.h.</var>
                ſimplicem, dabitur
                  <var>.a.h.</var>
                  <lb/>
                nempe
                  <var>.p.h.</var>
                minor numerus quæſitus.</s>
              </p>
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                <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a">
                  <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div96" type="math:theorem" level="3" n="45">
              <head xml:id="echoid-head61" xml:space="preserve">THEOREMA
                <num value="45">XLV</num>
              .</head>
              <p>
                <s xml:id="echoid-s390" xml:space="preserve">CVR volentes diuidere numerum propoſitum in duas eiuſmodi partes, vt pro
                  <lb/>
                ductum vnius in alteram, alteri numero propoſito æquetur, rectè dimidium
                  <lb/>
                primi dati numeri in ſeipſum multiplicant, ex quo quadrato ſecundum datum nu-
                  <lb/>
                merum detrahunt,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                radicem ſumunt, qua coniuncta vni dimidio primi nu-
                  <lb/>
                meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeſtabit.</s>
              </p>
              <p>
                <s xml:id="echoid-s391" xml:space="preserve">Exempli gratia, ſi numerus partiendus eſſet .34. alter verò numerus eſſet .64. cui
                  <lb/>
                productum vnius partis in alteram æquale eſſe deberet. </s>
                <s xml:id="echoid-s392" xml:space="preserve">Dimidium primi numeri, in
                  <lb/>
                ſeipſum multiplicaremus, cuius quadratum eſſet .289. de quo detracto ſecundo nu-
                  <lb/>
                mero nempe .64. remaneret .225. cuius quadrata radix nempe .15. coniuncta .17.
                  <lb/>
                dimidio .34. proferet .32. maiorem partem,
                  <reg norm="detractoque" type="simple">detractoq́;</reg>
                ex .17. ſupereſſet .2. pars
                  <lb/>
                inquam minor.</s>
              </p>
              <p>
                <s xml:id="echoid-s393" xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus propoſitus ſignificetur linea
                  <var>.a.d.</var>
                cu-
                  <lb/>
                ius dimidium
                  <var>.c.d.</var>
                cognitum erit, vnà etiam eius quadratum
                  <var>.c.f.</var>
                quo diuiſo per dia
                  <lb/>
                metrum
                  <var>.e.d.</var>
                ſupponantur partes ignotæ
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/>
                ipſius
                  <var>.a.d.</var>
                eſſe
                  <var>.a.b.</var>
                et
                  <var>.b.d.</var>
                & à puncto
                  <var>.b.</var>
                  <lb/>
                duci lineam
                  <var>.b.h.g.</var>
                parallelam
                  <var>.d.f.</var>
                et
                  <var>.m.
                    <lb/>
                  h.k.</var>
                parallelam
                  <var>.d.a.</var>
                extructa figura ſimi
                  <lb/>
                li figuræ quintæ ſecundi Eucli. </s>
                <s xml:id="echoid-s394" xml:space="preserve">quare da
                  <lb/>
                bitur
                  <reg norm="gnomon" type="context">gnomõ</reg>
                  <var>.l.d.g.</var>
                æqualis producto
                  <var>.b.
                    <lb/>
                  k.</var>
                & proinde cognitus, quo detracto è
                  <lb/>
                quadrato,
                  <var>c.f.</var>
                remanebit quadratum
                  <var>.g.l.</var>
                  <lb/>
                cuius radice æquali
                  <var>.c.b.</var>
                coniuncta
                  <var>.a.c.</var>
                  <lb/>
                & detracta ex
                  <var>.c.d.</var>
                partes
                  <var>.a.b.</var>
                et
                  <var>.b.d.</var>
                quæſitæ dabuntur.</s>
              </p>
              <div xml:id="echoid-div96" type="float" level="4" n="1">
                <figure xlink:label="fig-0041-02" xlink:href="fig-0041-02a">
                  <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div98" type="math:theorem" level="3" n="46">
              <head xml:id="echoid-head62" xml:space="preserve">THEOREMA
                <num value="46">XLVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s395" xml:space="preserve">CVR propoſitis tribus numeris, quorum prior in duas eiuſmodi partes diui-
                  <lb/>
                dendus ſit, ut mutuò diuiſæ, & per ſummam prouenientium diuiſo ſecundo
                  <lb/>
                numero, proueniens vltimum ſit æquale tertio numerorum propoſitorum. </s>
                <s xml:id="echoid-s396" xml:space="preserve">Conſul
                  <lb/>
                tiſsimum ſit ſecundum numerum per tertium diuidere, ex quo proueniens ſit ſum-
                  <lb/>
                ma prouenientium è duabus partibus mutuò diuiſis, quam ſummam ſi quis velit di-
                  <lb/>
                ſtinguere, rectè poſſit medio operationis
                  <reg norm="pręcedentis" type="context">pręcedẽtis</reg>
                theorematis
                  <reg norm="sumpta" type="context">sũpta</reg>
                vnitate ſuper
                  <lb/>
                ficiali pro ſecundo numero diſtinctis poſtmodum prouenientibus, rectè meo iudi-
                  <lb/>
                cio operabimur per
                  <reg norm="regulam" type="context">regulã</reg>
                de tribus (quod fuit ab antiquis prætermiſſum) Si dixe- </s>
              </p>
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