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-div89" type="math:theorem" level="3" n="41">
              <pb o="27" rhead="THEOREM. ARIT." n="39" file="0039" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0039"/>
              <p>
                <s xml:id="echoid-s364" xml:space="preserve">Quod vt ſpeculemus, conſideremus ſubſcriptam figuram, vigefiminoni theore-
                  <lb/>
                matis figuræ ſimilem, in qua numeri quæſiti duabus
                  <lb/>
                lineis directè coniunctis
                  <var>.q.g.</var>
                et
                  <var>.g.p.</var>
                fignificentur, ho
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/>
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                quadrata
                  <reg norm="erunt" type="context">erũt</reg>
                  <var>.r.c.</var>
                et
                  <var>.g.s.</var>
                  <reg norm="quorum" type="context">quorũ</reg>
                  <reg norm="summa" type="context">sũma</reg>
                  <reg norm="iterum" type="context">iterũ</reg>
                propo
                  <lb/>
                nitur, quare etiam cognita. </s>
                <s xml:id="echoid-s365" xml:space="preserve">
                  <reg norm="Differentia" type="context">Differẽtia</reg>
                autem
                  <reg norm="duorum" type="context">duorũ</reg>
                  <lb/>
                numerorum primo propofita fit
                  <var>.q.i.</var>
                eius verò qua-
                  <lb/>
                dratum
                  <var>.m.e.</var>
                quod cognitum eſt ex ſua radice
                  <var>.q.i</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s366" xml:space="preserve">quare gnomon
                  <var>.e.n.m.</var>
                ſimul cum quadrato minori
                  <var>.
                    <lb/>
                  g.s.</var>
                cognitus erit, quæ ſumma æqualis eſt duplo
                  <var>.g.r.</var>
                  <lb/>
                producto datorum numerorum. </s>
                <s xml:id="echoid-s367" xml:space="preserve">Itaque & ipſa
                  <var>.g.
                    <lb/>
                  r.</var>
                cognoſcetur, nunc ſi præcedentis theorematis ſpe-
                  <lb/>
                culationem in reliquis conſuluerimus propoſitum
                  <lb/>
                conſequemur.</s>
              </p>
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                <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a">
                  <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0039-01"/>
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            <div xml:id="echoid-div91" type="math:theorem" level="3" n="42">
              <head xml:id="echoid-head58" xml:space="preserve">THEOREMA
                <num value="42">XLII</num>
              .</head>
              <p>
                <s xml:id="echoid-s368" xml:space="preserve">ADhuc etiam & alia ratione idipſum conſequi poſſemus, non conſulto qua-
                  <lb/>
                drageſimo theoremate. </s>
                <s xml:id="echoid-s369" xml:space="preserve">Nam ſubtracto quadrato differentiæ, numeri primi
                  <lb/>
                (
                  <reg norm="inquam" type="context">inquã</reg>
                ) propoſiti, ex
                  <reg norm="summa" type="context">sũma</reg>
                duorum quadratorum, nempe ex ſecundo numero pro-
                  <lb/>
                poſito colligendum eſſet reſiduum in ſummam cum prædicto ſecundo numero, &
                  <lb/>
                ex ſumma hac deſumenda quadrata radix, quæ duorum numerorum ſumma erit,
                  <lb/>
                de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius
                  <lb/>
                dimidio addito primo numero propoſito, aut detracto minore inuento ex radice
                  <lb/>
                poſtremo inuenta, dabitur numerus maior, qui quæritur.</s>
              </p>
              <p>
                <s xml:id="echoid-s370" xml:space="preserve">Exempli gratia, cum ſuperfuerint .128. hæc ſi cum ſecundo numero
                  <reg norm="nempe" type="context">nẽpe</reg>
                .272.
                  <lb/>
                iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo
                  <lb/>
                propoſito, nempe .12. ſupererunt .8. quorum
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                erit .4. quo ex .20. detracto
                  <lb/>
                aut coniuncto .12. maior numerus orietur.</s>
              </p>
              <p>
                <s xml:id="echoid-s371" xml:space="preserve">Cuius rei contemplatio, præcedenti figura aperitur. </s>
                <s xml:id="echoid-s372" xml:space="preserve">Nam reſiduum detractionis
                  <lb/>
                quadrati
                  <var>.m.e.</var>
                ex ſumma
                  <reg norm="duorum" type="context">duorũ</reg>
                quadratorum
                  <var>.r.c.</var>
                et
                  <var>.g.s.</var>
                numerum præbet æqua-
                  <lb/>
                lem duobus ſupplementis
                  <var>.q.n.</var>
                et
                  <var>.n.u.</var>
                ex .8. ſecundi Euclidis. qui coniunctus duo-
                  <lb/>
                bus quadratis (quorum ſumma ſecundo propoſita fuit) cognitionem profert qua-
                  <lb/>
                drati
                  <var>.q.u.</var>
                & eius radicis
                  <var>.q.p.</var>
                de qua, detracto primo dato numero, ſcilicet
                  <var>.q.i.</var>
                ſu-
                  <lb/>
                pereſt
                  <var>.i.p.</var>
                cuius dimidium nempe
                  <var>.g.p.</var>
                minor eſt numerus qui quęritur; </s>
                <s xml:id="echoid-s373" xml:space="preserve">reſiduum
                  <lb/>
                verò totius
                  <var>.g.q.</var>
                maior ſcilicet.</s>
              </p>
            </div>
            <div xml:id="echoid-div92" type="math:theorem" level="3" n="43">
              <head xml:id="echoid-head59" xml:space="preserve">THEOREMA
                <num value="43">XLIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s374" xml:space="preserve">CVR ij, qui volunt duos numeros inuenire, quorum ſumma æqualis propo-
                  <lb/>
                fito alicui numero futura ſit, & ſumma quadratorum maior eorum produ-
                  <lb/>
                cto per quantitatem alterius propoſiti numeri, rectè dimidium primi dati numeri in
                  <lb/>
                ſeipſum multiplicant, quod quadratum ex
                  <reg norm="ſecundo" type="context">ſecũdo</reg>
                dato numero detrahunt, ſumunt­
                  <lb/>
                q́ue tertię partis refidui quadratam radicem, quam dimidio primi numeri coniun-
                  <lb/>
                gunt, ex quo maior numerus
                  <reg norm="duorum" type="context">duorũ</reg>
                  <reg norm="quæſitorum" type="context">quæſitorũ</reg>
                datur, quo ex toto primo detracto, ſu-
                  <lb/>
                pererit minor.</s>
              </p>
              <p>
                <s xml:id="echoid-s375" xml:space="preserve">Exempli gratia, propoſito numero .20. cui æquanda eſt ſumma duorum nume-
                  <lb/>
                rorum quæſitorum,
                  <reg norm="datoque" type="simple">datoq́;</reg>
                ſecundo numero .208. qui ſemper maior eſſe debet </s>
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