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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IO. BAPT. BENED.
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            <div xml:id="echoid-div130" type="math:theorem" level="3" n="64">
              <pb o="42" rhead="IO. BAPT. BENED." n="54" file="0054" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0054"/>
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            <div xml:id="echoid-div131" type="math:theorem" level="3" n="65">
              <head xml:id="echoid-head81" xml:space="preserve">THEOREMA
                <num value="65">LXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s557" xml:space="preserve">CVR propoſito numero in tres qualeſcunque partes diuiſo, ſi prima in
                  <lb/>
                tertiam multiplicetur, & huic producto, ſecundæ in primam productum
                  <lb/>
                coniungatur,
                  <reg norm="itemque" type="simple">itemq́;</reg>
                ſecundæ in tertiam, hæc ſumma duplicata æqualis ſit ſummæ
                  <lb/>
                productorum ſingularum in cæteras duas.</s>
              </p>
              <p>
                <s xml:id="echoid-s558" xml:space="preserve">Exempli gratia, ſi proponatur .20. diuiſus in tres partes nempe .12. 5. 3. multipli-
                  <lb/>
                cato primo .12. per .3. tertiam partem dabitur .36. ſecunda verò multiplicata per re
                  <lb/>
                liquas duas, hoc eſt .5. per .12. et .3. in primis dabitur .60. poſtea .15.
                  <reg norm="quorum" type="context">quorũ</reg>
                  <reg norm="trium" type="context">triũ</reg>
                pro
                  <lb/>
                ductorum ſumma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-
                  <lb/>
                citur ſummæ productorum ſingularum partium in reliquas duas, nempe ſummæ .60.
                  <lb/>
                36. 60. 15. 36. 15. hoc eſt ipſis .222.</s>
              </p>
              <p>
                <s xml:id="echoid-s559" xml:space="preserve">Cuius rei per ſe patet ſpeculatio, cum in his ſex vltimis productis, ſingula tria
                  <lb/>
                prima duplicentur.</s>
              </p>
            </div>
            <div xml:id="echoid-div132" type="math:theorem" level="3" n="66">
              <head xml:id="echoid-head82" xml:space="preserve">THEOREMA
                <num value="66">LXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s560" xml:space="preserve">CVR propoſito numero in .3. qualeſcunque partes diuiſo, ſi in reliquas duas ſin-
                  <lb/>
                gulæ multiplicentur, & hæc producta cum ſumma ſuorum quadratorum con-
                  <lb/>
                iungantur, tota ſumma hæc vltima æqualis erit quadrato totali propoſiti numeri.</s>
              </p>
              <p>
                <s xml:id="echoid-s561" xml:space="preserve">Exempli gratia, ſi fuerit idem numerus .20. in .3. partes diuiſus .12. 5. 3. </s>
                <s xml:id="echoid-s562" xml:space="preserve">Si .12. in
                  <lb/>
                5. et .3. producatur, ſumma productorum erit .96. at .5. in .12. et .3. erit .75. poſtmo-
                  <lb/>
                dum .3. in .12. et .5. erit .51. nempe in vniuerſum .222. quadratorum porrò ſumma
                  <lb/>
                erit .178 quæ coniuncta .222. dabit .400. quadratum ipſius .20.</s>
              </p>
              <p>
                <s xml:id="echoid-s563" xml:space="preserve">Erit autem huiuſce rei facillima ſpeculatio, ſi ſequentem figuram mente conce-
                  <lb/>
                perimus, in qua
                  <var>.a.b.</var>
                propoſitum numerum ſignificet, cuius partes diſtinctæ ſint me-
                  <lb/>
                dio
                  <var>.e.</var>
                et
                  <var>.c</var>
                . </s>
                <s xml:id="echoid-s564" xml:space="preserve">Ip ſum autem
                  <var>.q.b.</var>
                ſit quadratum
                  <lb/>
                totale parallelis
                  <var>.e.s.</var>
                et
                  <var>.c.x.</var>
                diuiſum, quæ qua
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0054-01a" xlink:href="fig-0054-01"/>
                dratum in triarectangula diuident, quorum
                  <lb/>
                primum erit
                  <var>.q.e.</var>
                compoſitum ex producto
                  <var>.a.
                    <lb/>
                  e.</var>
                in ſemetipſam, nempe quadratum
                  <var>.o.e.</var>
                &
                  <lb/>
                ex producto eiuſdem
                  <var>.a.e.</var>
                in
                  <var>.e.b.</var>
                quod erit re
                  <lb/>
                ctangulum
                  <var>.o.s.</var>
                ex quo tria rectangula
                  <var>.o.s.</var>
                et
                  <var>.
                    <lb/>
                  n.x.</var>
                et
                  <var>.t.u.</var>
                tria producta erunt ſingularum par
                  <lb/>
                tium in cæteras duas, et
                  <var>.e.o</var>
                :
                  <var>c.n</var>
                :
                  <var>b.t.</var>
                tria qua-
                  <lb/>
                drata erunt: </s>
                <s xml:id="echoid-s565" xml:space="preserve">quibus ſex quantitatibus quadra
                  <lb/>
                tum totale
                  <var>.q.b.</var>
                completur.</s>
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                <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a">
                  <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0054-01"/>
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            </div>
            <div xml:id="echoid-div134" type="math:theorem" level="3" n="67">
              <head xml:id="echoid-head83" xml:space="preserve">THEOREMA
                <num value="67">LXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s566" xml:space="preserve">
                  <emph style="sc">VEteres</emph>
                aliud quoque problema indefinitum propoſuerunt, quod tamen à
                  <lb/>
                nobis determinabitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s567" xml:space="preserve">Cur diuiſuri propoſitum numerum in duas eiuſmodi partes, vt mutuò diuiſis, &
                  <lb/>
                per ſummam prouenientium diuiſa ſumma qua dratorum partium, oriatur proue-
                  <lb/>
                niens alter numerus propoſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s568" xml:space="preserve">Propoſito deinde tertio quolibet numero diuidendo per ſingulas partes primi, </s>
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