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. ARITH.
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            <div xml:id="echoid-div98" type="math:theorem" level="3" n="46">
              <p>
                <s xml:id="echoid-s407" xml:space="preserve">
                  <pb o="31" rhead="THEOREM. ARITH." n="43" file="0043" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0043"/>
                ad vnitatem
                  <var>.o.i.</var>
                  <reg norm="permutandoque" type="simple">permutandoq́;</reg>
                  <var>.e.a.</var>
                ad
                  <var>.a.u.</var>
                ſicut
                  <var>.t.n.</var>
                ad
                  <var>.n.i.</var>
                & componendo
                  <var>.e.a.u.</var>
                  <lb/>
                ad
                  <var>a.u.</var>
                ſicut
                  <var>.t.n.i.</var>
                ad
                  <var>.n.i</var>
                : & euerſim
                  <var>.e.a.u.</var>
                ad
                  <var>.e.a.</var>
                vt
                  <var>.t.n.i.</var>
                ad
                  <var>.t.n</var>
                . </s>
                <s xml:id="echoid-s408" xml:space="preserve">Quare, ex .20. ſepti
                  <lb/>
                mi, recte vtimur regula de tribus. </s>
                <s xml:id="echoid-s409" xml:space="preserve">Idem & de altera parte dico, quamuis qui vnam
                  <lb/>
                teneat, alteram quo que habiturus ſit. </s>
                <s xml:id="echoid-s410" xml:space="preserve">Non mirum tamen ſi huiuſmodi problema
                  <lb/>
                ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</s>
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                <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a">
                  <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0042-01"/>
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            <div xml:id="echoid-div100" type="math:theorem" level="3" n="47">
              <head xml:id="echoid-head63" xml:space="preserve">THEOREMA
                <num value="47">XLVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s411" xml:space="preserve">CVR duobus numeris mutuó diuiſis, ſi per ſummam prouenientium, produ-
                  <lb/>
                ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-
                  <lb/>
                tn
                  <gap extent="2"/>
                m duorum numerorum æquale futurum ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s412" xml:space="preserve">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit
                  <num value="4">.
                    <lb/>
                  4.</num>
                integrorum cum quarta parte, qua ſumma multiplicata cum producto
                  <reg norm="primorum" type="context">primorũ</reg>
                  <lb/>
                numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-
                  <lb/>
                torum duorum numerorum æquantur.</s>
              </p>
              <p>
                <s xml:id="echoid-s413" xml:space="preserve">Hoc vt conſideremus, duo numeri partibus
                  <var>.a.e.</var>
                et
                  <var>.e.i.</var>
                in linea
                  <var>.a.i.</var>
                ſignificentur,
                  <lb/>
                quorum productum ſit
                  <var>.e.d.</var>
                &
                  <reg norm="quadratum" type="context">quadratũ</reg>
                ipſius
                  <var>.a.e.</var>
                ſit
                  <var>.e.p</var>
                : ipſius verò
                  <var>.e.i.</var>
                ſit
                  <var>.e.q.</var>
                pro-
                  <lb/>
                ueniens
                  <reg norm="autem" type="wordlist">aũt</reg>
                ex diuiſione
                  <var>.e.i.</var>
                per
                  <var>.a.e.</var>
                ſit
                  <var>.o.u.</var>
                proueniens
                  <reg norm="autem" type="wordlist">aũt</reg>
                  <var>.a.e.</var>
                per
                  <var>.e.i.</var>
                ſit
                  <var>.o.t.</var>
                quo-
                  <lb/>
                rum ſumma ſit
                  <var>.o.u.t.</var>
                tum productum
                  <var>.e.d</var>
                : linea
                  <var>.u.n.</var>
                ſignificetur ad angulum
                  <reg norm="rectum" type="context">rectũ</reg>
                  <lb/>
                coniuncta in puncto
                  <var>.u.</var>
                extremo ipſius
                  <var>.o.u.t.</var>
                productum
                  <reg norm="autem" type="wordlist">aũt</reg>
                  <var>.u.o.t.</var>
                in
                  <var>.u.n.</var>
                ſit
                  <var>.n.t</var>
                . </s>
                <s xml:id="echoid-s414" xml:space="preserve">Iam
                  <lb/>
                probandum nobis eſt
                  <var>.n.t.</var>
                æqualem eſſe ſummæ duorum quadratorum
                  <var>.q.e.p</var>
                . </s>
                <s xml:id="echoid-s415" xml:space="preserve">Quod
                  <lb/>
                ſingillatim probo, & aſſero productum
                  <var>.o.n.</var>
                æquale eſſe quadrato
                  <var>.q.e.</var>
                &
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.
                    <lb/>
                  s.t.</var>
                quadrato
                  <var>.e.p</var>
                . </s>
                <s xml:id="echoid-s416" xml:space="preserve">Nam ex .35. theoremate patet numerum
                  <var>.e.i.</var>
                medium eſſe
                  <reg norm="pro- portionalem" type="context">pro-
                    <lb/>
                  portionalẽ</reg>
                inter
                  <var>.e.d.</var>
                et
                  <var>.o.u</var>
                : cum numerus
                  <var>.e.i.</var>
                ex præſuppoſito ab
                  <var>.e.a.</var>
                multiplicetur
                  <lb/>
                & diuidatur, cuius multiplicationis produ-
                  <lb/>
                ctum eſt
                  <var>.d.e</var>
                : nempe
                  <var>.u.n.</var>
                & proueniens ex
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0043-01a" xlink:href="fig-0043-01"/>
                diuiſione eſt
                  <var>.o.u</var>
                : </s>
                <s xml:id="echoid-s417" xml:space="preserve">quare ex dicto theorema-
                  <lb/>
                te
                  <var>.e.i.</var>
                media proportionalis eſt inter
                  <var>.u.n.</var>
                et
                  <var>.
                    <lb/>
                  u.o</var>
                . </s>
                <s xml:id="echoid-s418" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                productum
                  <var>.o.n.</var>
                æquale eſt qua-
                  <lb/>
                drato
                  <var>.e.q.</var>
                ex .16. ſexti vel .20. ſeptimi. </s>
                <s xml:id="echoid-s419" xml:space="preserve">Idem
                  <lb/>
                dico de producto
                  <var>.s.t.</var>
                  <reg norm="nempe" type="context">nẽpe</reg>
                æquale eſſe qua-
                  <lb/>
                drato
                  <var>.e.p.</var>
                quandoquidem numerus
                  <var>.a.e.</var>
                ab
                  <lb/>
                  <var>e.i.</var>
                multiplicatur ac diuiditur, cuius multi-
                  <lb/>
                plicationis productum eſt
                  <var>.d.e.</var>
                nempe
                  <var>o.s.</var>
                &
                  <lb/>
                proueniens ex diuiſione
                  <var>.o.t</var>
                : </s>
                <s xml:id="echoid-s420" xml:space="preserve">inter quæ ex .
                  <lb/>
                35. theoremate
                  <var>.a.e.</var>
                media proportionalis
                  <lb/>
                eſt. </s>
                <s xml:id="echoid-s421" xml:space="preserve">Quare ex allatis propoſitionibus
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.s.t.</var>
                æquale eſt quadrato
                  <var>.e.p.</var>
                ſed
                  <reg norm="totum" type="context">totũ</reg>
                  <lb/>
                productum
                  <var>.n.t.</var>
                ſumma eſt duorum productorum
                  <var>.o.n.</var>
                et
                  <var>.s.t.</var>
                ex prima ſecundi Eucli.
                  <lb/>
                </s>
                <s xml:id="echoid-s422" xml:space="preserve">Itaque verum eſſe quod dictum eſt, conſequitur.</s>
              </p>
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                <figure xlink:label="fig-0043-01" xlink:href="fig-0043-01a">
                  <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0043-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div102" type="math:theorem" level="3" n="48">
              <head xml:id="echoid-head64" xml:space="preserve">THEOREMA
                <num value="48">XLVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s423" xml:space="preserve">CVR ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium,
                  <lb/>
                per minorem diuidat,
                  <reg norm="maioremque" type="simple">maioremq́;</reg>
                per proueniens multiplicet, productum,
                  <lb/>
                  <reg norm="summæ" type="context">sũmæ</reg>
                ipſius maioris cum eodem proueniente æquale erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s424" xml:space="preserve">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-
                  <lb/>
                to per proueniens, ipſo nempe .10: </s>
                <s xml:id="echoid-s425" xml:space="preserve">datur productum .11. cum nona parte, tantum ſci­ </s>
              </p>
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